npc and finitely presented non-positive curvature and ... · finitely presented groups from...
TRANSCRIPT
![Page 1: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/1.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Non-positive curvature and complexityfor finitely presented groups
Martin R Bridson
Department of MathematicsImperial College London
ICM Madrid, 24 August 2006.
![Page 2: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/2.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline
Finitely Presented GroupsFrom presentations to geometryThe 2 Strands of Geometric Group TheoryThe universe of finitely presented groups
Non-Positive Curvaturelocal curvature, CAT(0)NPC groups
Isoperimetric functionsDehn functionsIsoperimetric spectra
Drawing in and Reaching outSubdirect products of hyperbolic groupsSolution of Grothendieck’s Problems
![Page 3: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/3.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Finitely presented groups
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉 ≡ P
a1
a2
a3a4
an
r1 r2 r3
r4 rm
Figure: The standard 2-complex K (P)
![Page 4: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/4.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Finitely presented groups
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉 ≡ P
a1
a2
a3a4
an
r1 r2 r3
r4 rm
Figure: The standard 2-complex K (P)
![Page 5: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/5.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The group springing into action
=a
a
a
aaaa
a
b bb
b
b
b
b
Figure: The 2-complex and Cayley graph for 〈a, b | [a, b]〉
![Page 6: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/6.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The first strand in Geometric Group Theory
Strand 1: Study (and manufacture) group actions onspaces in order to elucidate the structure of both thegroups and the spaces.
One may prefer discrete cocompact actions by isometriesbut . . .sometimes weaken admission criteria to obtain a morediverse class of groups,sometimes demand more structure to narrow the focuson groups and spaces of exceptional character.
![Page 7: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/7.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The first strand in Geometric Group Theory
Strand 1: Study (and manufacture) group actions onspaces in order to elucidate the structure of both thegroups and the spaces.
One may prefer discrete cocompact actions by isometriesbut . . .sometimes weaken admission criteria to obtain a morediverse class of groups,sometimes demand more structure to narrow the focuson groups and spaces of exceptional character.
![Page 8: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/8.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The first strand in Geometric Group Theory
Strand 1: Study (and manufacture) group actions onspaces in order to elucidate the structure of both thegroups and the spaces.
One may prefer discrete cocompact actions by isometriesbut . . .sometimes weaken admission criteria to obtain a morediverse class of groups,sometimes demand more structure to narrow the focuson groups and spaces of exceptional character.
![Page 9: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/9.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Groups as geometric objects (Gromov)
Study finitely generated groups as geometric objects intheir own right, via their intrinsic geometry.
Γ = 〈a1, . . . , an | r1, r2, . . . 〉Word Metric:
d(γ1, γ2) = min{|w | : w ∈ F (A), w Γ= γ−1
1 γ2}.
Cayley Graph (1878) = K (P)(1)
� Word metric and Cayley graph are independent ofgenerating set, up to quasi-isometry.
� Thus one is particularly interested in properties ofgroups and spaces invariant under quasi-isometry.
� Large-scale (coarse) geometry and topology
![Page 10: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/10.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Groups as geometric objects (Gromov)
Study finitely generated groups as geometric objects intheir own right, via their intrinsic geometry.
Γ = 〈a1, . . . , an | r1, r2, . . . 〉Word Metric:
d(γ1, γ2) = min{|w | : w ∈ F (A), w Γ= γ−1
1 γ2}.
Cayley Graph (1878) = K (P)(1)
� Word metric and Cayley graph are independent ofgenerating set, up to quasi-isometry.
� Thus one is particularly interested in properties ofgroups and spaces invariant under quasi-isometry.
� Large-scale (coarse) geometry and topology
![Page 11: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/11.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Groups as geometric objects (Gromov)
Study finitely generated groups as geometric objects intheir own right, via their intrinsic geometry.
Γ = 〈a1, . . . , an | r1, r2, . . . 〉Word Metric:
d(γ1, γ2) = min{|w | : w ∈ F (A), w Γ= γ−1
1 γ2}.
Cayley Graph (1878) = K (P)(1)
� Word metric and Cayley graph are independent ofgenerating set, up to quasi-isometry.
� Thus one is particularly interested in properties ofgroups and spaces invariant under quasi-isometry.
� Large-scale (coarse) geometry and topology
![Page 12: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/12.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Groups as geometric objects (Gromov)
Study finitely generated groups as geometric objects intheir own right, via their intrinsic geometry.
Γ = 〈a1, . . . , an | r1, r2, . . . 〉Word Metric:
d(γ1, γ2) = min{|w | : w ∈ F (A), w Γ= γ−1
1 γ2}.
Cayley Graph (1878) = K (P)(1)
� Word metric and Cayley graph are independent ofgenerating set, up to quasi-isometry.
� Thus one is particularly interested in properties ofgroups and spaces invariant under quasi-isometry.
� Large-scale (coarse) geometry and topology
![Page 13: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/13.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Combinatorial Group Theory (Dehn 1912)
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉
“The general discontinuous group is given [as above].There are above all three fundamental problems.
� The identity [word] problem� The transformation [conjugacy] problem� The isomorphism problem
[. . .] One is already led to them by necessity with work intopology. Each knotted space curve, in order to becompletely understood, demands the solution of thethree"
Higman Embedding (1961): Every recursively presentedgroup is a subgroup of a finitely presented group.
![Page 14: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/14.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Combinatorial Group Theory (Dehn 1912)
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉
“The general discontinuous group is given [as above].There are above all three fundamental problems.
� The identity [word] problem� The transformation [conjugacy] problem� The isomorphism problem
[. . .] One is already led to them by necessity with work intopology. Each knotted space curve, in order to becompletely understood, demands the solution of thethree"
Higman Embedding (1961): Every recursively presentedgroup is a subgroup of a finitely presented group.
![Page 15: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/15.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Combinatorial Group Theory (Dehn 1912)
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉
“The general discontinuous group is given [as above].There are above all three fundamental problems.
� The identity [word] problem� The transformation [conjugacy] problem� The isomorphism problem
[. . .] One is already led to them by necessity with work intopology. Each knotted space curve, in order to becompletely understood, demands the solution of thethree"
Higman Embedding (1961): Every recursively presentedgroup is a subgroup of a finitely presented group.
![Page 16: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/16.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Combinatorial Group Theory (Dehn 1912)
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉
“The general discontinuous group is given [as above].There are above all three fundamental problems.
� The identity [word] problem� The transformation [conjugacy] problem� The isomorphism problem
[. . .] One is already led to them by necessity with work intopology. Each knotted space curve, in order to becompletely understood, demands the solution of thethree"
Higman Embedding (1961): Every recursively presentedgroup is a subgroup of a finitely presented group.
![Page 17: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/17.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Combinatorial Group Theory (Dehn 1912)
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉
“The general discontinuous group is given [as above].There are above all three fundamental problems.
� The identity [word] problem� The transformation [conjugacy] problem� The isomorphism problem
[. . .] One is already led to them by necessity with work intopology. Each knotted space curve, in order to becompletely understood, demands the solution of thethree"
Higman Embedding (1961): Every recursively presentedgroup is a subgroup of a finitely presented group.
![Page 18: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/18.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Combinatorial Group Theory (Dehn 1912)
Γ ∼= 〈a1, . . . , an | r1, . . . , rm〉
“The general discontinuous group is given [as above].There are above all three fundamental problems.
� The identity [word] problem� The transformation [conjugacy] problem� The isomorphism problem
[. . .] One is already led to them by necessity with work intopology. Each knotted space curve, in order to becompletely understood, demands the solution of thethree"
Higman Embedding (1961): Every recursively presentedgroup is a subgroup of a finitely presented group.
![Page 19: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/19.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The universe of finitely presented groups
1?
?
?
?
??
?
?Z
FEF
HYP
C0SH
AUTO
IP(2)
COMB
ASYN
AB
NILP
PC
SOL
EA
L
vNTAMENABLE
HYPERBOLIC
NPC
HIC ABUNDANTLEONES
![Page 20: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/20.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Gromov’s hyperbolic groups
If Γ is hyperbolic then it
• acts properly, cocompactly on a contractible complex
• has only finitely many conjugacy classes of finitesubgroups and its abelian subgroups are virtually cyclic
• striking algorithmic properties: the set of geodesicwords for Γ (wrt any finite gen set) is a regular language:∃ finite state automaton recognising words labellinggeodesics in Cayley graph; hyperbolic groups areautomatic.
• Rapidly-solvable word and conjugacy problems.
• [Sela] The isomorphism problem is solvable amongtorsion-free hyperbolic groups.
![Page 21: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/21.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Gromov’s hyperbolic groups
If Γ is hyperbolic then it
• acts properly, cocompactly on a contractible complex
• has only finitely many conjugacy classes of finitesubgroups and its abelian subgroups are virtually cyclic
• striking algorithmic properties: the set of geodesicwords for Γ (wrt any finite gen set) is a regular language:∃ finite state automaton recognising words labellinggeodesics in Cayley graph; hyperbolic groups areautomatic.
• Rapidly-solvable word and conjugacy problems.
• [Sela] The isomorphism problem is solvable amongtorsion-free hyperbolic groups.
![Page 22: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/22.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
CAT(0) and CAT(-1) conditions
A.D. Alexandrov
d d�
d
x y
X
d
x y
E2
Figure: The CAT(0) inequality
Local-to-global: If X is complete and every point has aneighbourhood in which triangles satisfy this inequality,then in X all triangles satisfy this inequality.
![Page 23: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/23.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
CAT(0) and CAT(-1) conditions
A.D. Alexandrov
d d�
d
x y
X
d
x y
E2
Figure: The CAT(0) inequality
Local-to-global: If X is complete and every point has aneighbourhood in which triangles satisfy this inequality,then in X all triangles satisfy this inequality.
![Page 24: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/24.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
CAT(0) spaces
Metric spaces of non-positive curvature, Bridson-HaefligerGrund. Math. Wiss. 319
• connections with many branches of mathematics
• local-to-global phenomena
• rigidity
• complexes of groups (à la Haefliger)
• connections with 3-dimensional geometry
• combination theorems; verifiability
• A great deal one can say about the structure of groupsthat act by isometries on CAT(0) and CAT(−1) spaces
![Page 25: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/25.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Negative curvature and hyperbolic groups
Identify key robust feature of CAT(−1) spaces X
If Γ acts geometrically on X (basepoint p), articulate whatremains of the feature when it is pulled-back via theΓ-equivariant quasi-isometry γ �→ γ.p (fixed p ∈ X ).
��
Figure: The slim triangles condition
![Page 26: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/26.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Negative curvature and hyperbolic groups
Identify key robust feature of CAT(−1) spaces X
If Γ acts geometrically on X (basepoint p), articulate whatremains of the feature when it is pulled-back via theΓ-equivariant quasi-isometry γ �→ γ.p (fixed p ∈ X ).
��
Figure: The slim triangles condition
![Page 27: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/27.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Coarse convexity
< k
1
γ
γa
σ
σγa
γ
Figure: The fellow-traveller property
![Page 28: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/28.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The universe of finitely presented groups
1?
?
?
?
??
?
?Z
FEF
HYP
C0SH
AUTO
IP(2)
COMB
ASYN
AB
NILP
PC
SOL
EA
L
vNTAMENABLE
HYPERBOLIC
NPC
HIC ABUNDANTLEONES
![Page 29: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/29.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Discriminating among NPC Groups [B, c.’00]
Theorem∃ combable groups that are not bicombable or automatic.
Hierarchy of formal languages, Reg ⊂ CF ⊂ Ind.
Theorem∃ Ind-combable groups that are not Reg-combable(automatic); some have quadratic Dehn functions, somecubic.
Theorem∃ combable groups with unsolvable conjugacy problem.
TheoremIsomorphism problem unsolvable for combable groups.
![Page 30: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/30.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Discriminating among NPC Groups [B, c.’00]
Theorem∃ combable groups that are not bicombable or automatic.
Hierarchy of formal languages, Reg ⊂ CF ⊂ Ind.
Theorem∃ Ind-combable groups that are not Reg-combable(automatic); some have quadratic Dehn functions, somecubic.
Theorem∃ combable groups with unsolvable conjugacy problem.
TheoremIsomorphism problem unsolvable for combable groups.
![Page 31: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/31.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Discriminating among NPC Groups [B, c.’00]
Theorem∃ combable groups that are not bicombable or automatic.
Hierarchy of formal languages, Reg ⊂ CF ⊂ Ind.
Theorem∃ Ind-combable groups that are not Reg-combable(automatic); some have quadratic Dehn functions, somecubic.
Theorem∃ combable groups with unsolvable conjugacy problem.
TheoremIsomorphism problem unsolvable for combable groups.
![Page 32: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/32.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Discriminating among NPC Groups [B, c.’00]
Theorem∃ combable groups that are not bicombable or automatic.
Hierarchy of formal languages, Reg ⊂ CF ⊂ Ind.
Theorem∃ Ind-combable groups that are not Reg-combable(automatic); some have quadratic Dehn functions, somecubic.
Theorem∃ combable groups with unsolvable conjugacy problem.
TheoremIsomorphism problem unsolvable for combable groups.
![Page 33: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/33.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Dehn functions of groups
Finitely presented Γ = 〈A | R〉, word w ∈ F (A) withw = 1 in Γ
Area(w):= min
N : w =
N∏j=1
ujrju−1j freely, rj ∈ R±1
The Dehn function δ(n) of the presentation is
δ(n) = max{Area(w)| w ∈ ker(F (A) → Γ), |w | ≤ n }
where |w | denotes the length of the word w .
![Page 34: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/34.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Dehn functions of groups
Finitely presented Γ = 〈A | R〉, word w ∈ F (A) withw = 1 in Γ
Area(w):= min
N : w =
N∏j=1
ujrju−1j freely, rj ∈ R±1
The Dehn function δ(n) of the presentation is
δ(n) = max{Area(w)| w ∈ ker(F (A) → Γ), |w | ≤ n }
where |w | denotes the length of the word w .
![Page 35: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/35.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The Filling Theorem
[Gromov]W smooth, complete, Riemannian manifold.
• c : S1 → W null-homotopic, rectifiable loop
FArea(c)= inf{Area(F ) | F : D2 → W , F |∂D2 = c}.
• The 2-diml isoperimetric function [0,∞) → [0,∞) is
FillM0 (r):= sup{FArea(c) | c : S1 → M, l(c) ≤ r}.
• Filling Theorem: ∀M closed, FillM0 (x) δπ1M(x).
![Page 36: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/36.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The Filling Theorem
[Gromov]W smooth, complete, Riemannian manifold.
• c : S1 → W null-homotopic, rectifiable loop
FArea(c)= inf{Area(F ) | F : D2 → W , F |∂D2 = c}.
• The 2-diml isoperimetric function [0,∞) → [0,∞) is
FillM0 (r):= sup{FArea(c) | c : S1 → M, l(c) ≤ r}.
• Filling Theorem: ∀M closed, FillM0 (x) δπ1M(x).
![Page 37: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/37.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The Filling Theorem
[Gromov]W smooth, complete, Riemannian manifold.
• c : S1 → W null-homotopic, rectifiable loop
FArea(c)= inf{Area(F ) | F : D2 → W , F |∂D2 = c}.
• The 2-diml isoperimetric function [0,∞) → [0,∞) is
FillM0 (r):= sup{FArea(c) | c : S1 → M, l(c) ≤ r}.
• Filling Theorem: ∀M closed, FillM0 (x) δπ1M(x).
![Page 38: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/38.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The isoperimetric spectrum
IP = {α | nα Dehn function} ⊆ [1,∞)
• Gromov (Bowditch, Papasoglu, Olshanskii,...):(1, 2) ∩ IP = ∅ and α = 1 are the hyperbolic groups.
• At α = 2 one has a rich and diverse class of groups.
•N ⊆ IP [Baumslag-Miller-Short, B-Pittet, Gromov]
• Bridson (’94): ∃ infinitely many non-integers in IP.
• Brady-B (’98): ∀p ≥ q integers, 2 log2(2p/q) ∈ IP.
• Sapir-Birget-Rips (’98): Deeper analysis δ(x) � x4.
• B,B,Forester,Shankar (’04) Q ∩ [2,∞) ⊆ IP.
Theorem (B-Brady ’98)
IP = {1} ∪ [2,∞).
![Page 39: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/39.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Basic snowflake groups [Brady-B]
G = 〈a1, a2, c, s1, s2 |a1a2 = c = a2a1, s−1i ar
i si = c〉
Figure: Half of snowflake diagram and dual tree
If k is the radius of the dual tree,
• Growth of the base of central ∆ is r k
• Area of central ∆ is roughly square of the base.
• Length of boundary is roughly 2k .
• Area ≥ (r k )2 = (2k )2 log2(r) = |∂|2 log2(r)
![Page 40: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/40.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Basic snowflake groups [Brady-B]
G = 〈a1, a2, c, s1, s2 |a1a2 = c = a2a1, s−1i ar
i si = c〉
Figure: Half of snowflake diagram and dual tree
If k is the radius of the dual tree,
• Growth of the base of central ∆ is r k
• Area of central ∆ is roughly square of the base.
• Length of boundary is roughly 2k .
• Area ≥ (r k )2 = (2k )2 log2(r) = |∂|2 log2(r)
![Page 41: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/41.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Basic snowflake groups [Brady-B]
G = 〈a1, a2, c, s1, s2 |a1a2 = c = a2a1, s−1i ar
i si = c〉
Figure: Half of snowflake diagram and dual tree
If k is the radius of the dual tree,
• Growth of the base of central ∆ is r k
• Area of central ∆ is roughly square of the base.
• Length of boundary is roughly 2k .
• Area ≥ (r k )2 = (2k )2 log2(r) = |∂|2 log2(r)
![Page 42: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/42.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
General BBFS construction based onPerron-Frobenius theory
Figure: A snowflake disk based on the matrix P = ( 12
11 )
![Page 43: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/43.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Take iterated mapping tori by homotheties
Figure: Snowflake spheres
![Page 44: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/44.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Isoperimetric spectra
1 2 332
43
54
65
IP
IP(2)
IP(3)
IP(4)
IP(5)
↖ ×Z ↑ Σ
Figure: Complete picture of IP, partial for IP(k).
IP(k) = {α | nα k-diml Dehn function}
![Page 45: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/45.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Subdirect products of hyperbolic groups
Tame: Finitely presented subdirect products of freegroups, surface groups and limit groups are remarkablytame, for example:
Theorem (B,Howie,Miller,Short)If the Γi are limit groups and S ⊂ Γ1 × · · · × Γn is of typeFPn, then S is virtually a direct product of limit groups.
Wild: Finitely presented subgroups of more generalhyperbolic groups can be utterly wild.
![Page 46: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/46.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
An embracing algorithm (Rips & 1-2-3 Thm)
∃ algorithm with input a finite aspherical presentation Qand output a FINITE presentation of P ⊂ H × H, with Hhyperbolic.
P := {(γ1, γ2) | p(γ1) = p(γ2)} ⊂ H × H
is the fibre-product associated to s.e.s.
1 → N → Hp→ Q → 1
with N fin gen, H 2-diml hyperbolic, Q = |Q|.
“1-2-3 Thm" refers to fact that N, H and Q are of typeF1,F2 and F3 respectively. [Baumslag, B, Miller, Short]
Refinements (B-Haefliger, Wise) place more stringentconditions on H, e.g. locally CAT(−1) or residually finite.
![Page 47: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/47.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
An embracing algorithm (Rips & 1-2-3 Thm)
∃ algorithm with input a finite aspherical presentation Qand output a FINITE presentation of P ⊂ H × H, with Hhyperbolic.
P := {(γ1, γ2) | p(γ1) = p(γ2)} ⊂ H × H
is the fibre-product associated to s.e.s.
1 → N → Hp→ Q → 1
with N fin gen, H 2-diml hyperbolic, Q = |Q|.
“1-2-3 Thm" refers to fact that N, H and Q are of typeF1,F2 and F3 respectively. [Baumslag, B, Miller, Short]
Refinements (B-Haefliger, Wise) place more stringentconditions on H, e.g. locally CAT(−1) or residually finite.
![Page 48: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/48.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
An embracing algorithm (Rips & 1-2-3 Thm)
∃ algorithm with input a finite aspherical presentation Qand output a FINITE presentation of P ⊂ H × H, with Hhyperbolic.
P := {(γ1, γ2) | p(γ1) = p(γ2)} ⊂ H × H
is the fibre-product associated to s.e.s.
1 → N → Hp→ Q → 1
with N fin gen, H 2-diml hyperbolic, Q = |Q|.
“1-2-3 Thm" refers to fact that N, H and Q are of typeF1,F2 and F3 respectively. [Baumslag, B, Miller, Short]
Refinements (B-Haefliger, Wise) place more stringentconditions on H, e.g. locally CAT(−1) or residually finite.
![Page 49: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/49.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
An embracing algorithm (Rips & 1-2-3 Thm)
∃ algorithm with input a finite aspherical presentation Qand output a FINITE presentation of P ⊂ H × H, with Hhyperbolic.
P := {(γ1, γ2) | p(γ1) = p(γ2)} ⊂ H × H
is the fibre-product associated to s.e.s.
1 → N → Hp→ Q → 1
with N fin gen, H 2-diml hyperbolic, Q = |Q|.
“1-2-3 Thm" refers to fact that N, H and Q are of typeF1,F2 and F3 respectively. [Baumslag, B, Miller, Short]
Refinements (B-Haefliger, Wise) place more stringentconditions on H, e.g. locally CAT(−1) or residually finite.
![Page 50: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/50.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
An embracing algorithm (Rips & 1-2-3 Thm)
∃ algorithm with input a finite aspherical presentation Qand output a FINITE presentation of P ⊂ H × H, with Hhyperbolic.
P := {(γ1, γ2) | p(γ1) = p(γ2)} ⊂ H × H
is the fibre-product associated to s.e.s.
1 → N → Hp→ Q → 1
with N fin gen, H 2-diml hyperbolic, Q = |Q|.
“1-2-3 Thm" refers to fact that N, H and Q are of typeF1,F2 and F3 respectively. [Baumslag, B, Miller, Short]
Refinements (B-Haefliger, Wise) place more stringentconditions on H, e.g. locally CAT(−1) or residually finite.
![Page 51: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/51.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
An embracing algorithm (Rips & 1-2-3 Thm)
∃ algorithm with input a finite aspherical presentation Qand output a FINITE presentation of P ⊂ H × H, with Hhyperbolic.
P := {(γ1, γ2) | p(γ1) = p(γ2)} ⊂ H × H
is the fibre-product associated to s.e.s.
1 → N → Hp→ Q → 1
with N fin gen, H 2-diml hyperbolic, Q = |Q|.
“1-2-3 Thm" refers to fact that N, H and Q are of typeF1,F2 and F3 respectively. [Baumslag, B, Miller, Short]
Refinements (B-Haefliger, Wise) place more stringentconditions on H, e.g. locally CAT(−1) or residually finite.
![Page 52: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/52.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Grothendieck’s Question (1970)
A �= 0 a commutative ring, Γ a finitely generated group,RepA(Γ) the category of Γ-actions on fin. pres. A-modules.Any homomorphism u : Γ1 → Γ2 of groups induces
u∗A : RepA(Γ2) → RepA(Γ1).
Theorem (G, 1970)If u : Γ1 → Γ2 is a homomorphism of finitely generatedgroups, u∗
A is an equivalence of categories if and only ifu : Γ1 → Γ2 is an isomorphism of profinite groups.
Question (G, 1970): If Γ1 and Γ2 are finitely presentedand residually finite, is u : Γ1 → Γ2 an isomorphism?
![Page 53: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/53.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Grothendieck’s Question (1970)
A �= 0 a commutative ring, Γ a finitely generated group,RepA(Γ) the category of Γ-actions on fin. pres. A-modules.Any homomorphism u : Γ1 → Γ2 of groups induces
u∗A : RepA(Γ2) → RepA(Γ1).
Theorem (G, 1970)If u : Γ1 → Γ2 is a homomorphism of finitely generatedgroups, u∗
A is an equivalence of categories if and only ifu : Γ1 → Γ2 is an isomorphism of profinite groups.
Question (G, 1970): If Γ1 and Γ2 are finitely presentedand residually finite, is u : Γ1 → Γ2 an isomorphism?
![Page 54: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/54.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Grothendieck’s Question (1970)
A �= 0 a commutative ring, Γ a finitely generated group,RepA(Γ) the category of Γ-actions on fin. pres. A-modules.Any homomorphism u : Γ1 → Γ2 of groups induces
u∗A : RepA(Γ2) → RepA(Γ1).
Theorem (G, 1970)If u : Γ1 → Γ2 is a homomorphism of finitely generatedgroups, u∗
A is an equivalence of categories if and only ifu : Γ1 → Γ2 is an isomorphism of profinite groups.
Question (G, 1970): If Γ1 and Γ2 are finitely presentedand residually finite, is u : Γ1 → Γ2 an isomorphism?
![Page 55: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/55.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Grothendieck’s Question (1970)
A �= 0 a commutative ring, Γ a finitely generated group,RepA(Γ) the category of Γ-actions on fin. pres. A-modules.Any homomorphism u : Γ1 → Γ2 of groups induces
u∗A : RepA(Γ2) → RepA(Γ1).
Theorem (G, 1970)If u : Γ1 → Γ2 is a homomorphism of finitely generatedgroups, u∗
A is an equivalence of categories if and only ifu : Γ1 → Γ2 is an isomorphism of profinite groups.
Question (G, 1970): If Γ1 and Γ2 are finitely presentedand residually finite, is u : Γ1 → Γ2 an isomorphism?
![Page 56: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/56.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Solution of Grothendieck’s Problem
Qu: If Γ1 and Γ2 are finitely presented and residuallyfinite, must u : Γ1 → Γ2 be an isomorphism if u : Γ1 → Γ2
is an isomorphism?
Grothendieck proved that the answer is yes in manycases, e.g. arithmetic groups. Platonov-Tavgen (laterBass–Lubotzky, Pyber) proved answer no for finitelygenerated groups in general.
Theorem (B-Grunewald, 2003)∃ residually finite, hyperbolic groups H and finitelypresented subgroups P ↪→ Γ := H × H of infinite index,such that P is not abstractly isomorphic to Γ, but theinclusion u : P ↪→ Γ induces an isomorphism u : P → Γ.
![Page 57: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/57.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Solution of Grothendieck’s Problem
Qu: If Γ1 and Γ2 are finitely presented and residuallyfinite, must u : Γ1 → Γ2 be an isomorphism if u : Γ1 → Γ2
is an isomorphism?
Grothendieck proved that the answer is yes in manycases, e.g. arithmetic groups. Platonov-Tavgen (laterBass–Lubotzky, Pyber) proved answer no for finitelygenerated groups in general.
Theorem (B-Grunewald, 2003)∃ residually finite, hyperbolic groups H and finitelypresented subgroups P ↪→ Γ := H × H of infinite index,such that P is not abstractly isomorphic to Γ, but theinclusion u : P ↪→ Γ induces an isomorphism u : P → Γ.
![Page 58: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/58.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Solution of Grothendieck’s Problem
Qu: If Γ1 and Γ2 are finitely presented and residuallyfinite, must u : Γ1 → Γ2 be an isomorphism if u : Γ1 → Γ2
is an isomorphism?
Grothendieck proved that the answer is yes in manycases, e.g. arithmetic groups. Platonov-Tavgen (laterBass–Lubotzky, Pyber) proved answer no for finitelygenerated groups in general.
Theorem (B-Grunewald, 2003)∃ residually finite, hyperbolic groups H and finitelypresented subgroups P ↪→ Γ := H × H of infinite index,such that P is not abstractly isomorphic to Γ, but theinclusion u : P ↪→ Γ induces an isomorphism u : P → Γ.
![Page 59: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/59.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 60: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/60.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 61: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/61.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 62: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/62.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 63: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/63.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 64: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/64.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 65: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/65.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Outline of the proof
Suppose Q has no finite quotients.
1 → N → H → Q → 1
Lemma. H2(Q, Z) = 0 =⇒ N → H is iso.
� Build Q with aspherical presentation, no finitequotients and H2(Q, Z) = 0〈a1, a2, b1, b2 | a−1
1 ap2a1a−p−1
2 , b−11 bp
2b1b−p−12 ,
. a−11 [b2, b−1
1 b2b1], b−11 [a2, a−1
1 a2a1]〉
� Apply the embracing algorithm: P ⊂ H × H� Extend the lemma to deal with fibre products.
![Page 66: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/66.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
Theorem (B-Grunewald, 2003)∃ residually finite, hyperbolic groups H and finitelypresented subgroups P ↪→ Γ := H × H of infinite index,such that P is not abstractly isomorphic to Γ, but theinclusion u : P ↪→ Γ induces an isomorphism u : P → Γ.
![Page 67: NPC and Finitely Presented Non-positive curvature and ... · Finitely Presented Groups From presentations to geometry The 2 Strands of Geometric Group Theory ... Study finitely generated](https://reader033.vdocuments.mx/reader033/viewer/2022052518/5f0664047e708231d417c217/html5/thumbnails/67.jpg)
NPC andcomplexity for f.p.
groups
Martin R Bridson
Finitely PresentedGroupsFrom presentations togeometry
The 2 Strands of GeometricGroup Theory
The universe of finitelypresented groups
Non-PositiveCurvaturelocal curvature, CAT(0)
NPC groups
IsoperimetricfunctionsDehn functions
Isoperimetric spectra
Drawing in andReaching outSubdirect products ofhyperbolic groups
Solution of Grothendieck’sProblems
The universe of finitely presented groups
1?
?
?
?
??
?
?Z
FEF
HYP
C0SH
AUTO
IP(2)
COMB
ASYN
AB
NILP
PC
SOL
EA
L
vNTAMENABLE
HYPERBOLIC
NPC
HIC ABUNDANTLEONES