commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · phillip wesolek...
TRANSCRIPT
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Commensurated subgroups of finitely generatedbranch groups
Phillip Wesolek
Binghamton University
Permutation Groups, BIRS
Phillip Wesolek Commensurated subgroups 11/15/2016 1 / 18
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DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G.
The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
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DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
![Page 4: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/4.jpg)
DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups
2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
![Page 5: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/5.jpg)
DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups
3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
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DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.
4 SL3(Z) in SL3(Z[1p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
![Page 7: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/7.jpg)
DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
![Page 8: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/8.jpg)
DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V
6 Any compact open subgroup of a totally disconnected locallycompact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
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DefinitionA subgroup K of a group G is commensurated if |K : K ∩ gKg−1| <∞for all g ∈ G. The pair (G,K ) is sometimes called a Hecke pair.
Examples1 Normal subgroups2 Finite subgroups3 Point stabilizers of transitive sub-degree finite permutation groups.4 SL3(Z) in SL3(Z[1
p ])
5 Vaut in Thompson’s group V6 Any compact open subgroup of a totally disconnected locally
compact (t.d.l.c.) group
Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18
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Say (G,K ) is a Hecke pair.
The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication.
This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ).
The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair.
The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.
2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.
3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Say (G,K ) is a Hecke pair. The group G acts on the set of left cosetsG/K by left multiplication. This induces a permutation representationσ : G→ Sym(G/K ). The group Sym(G/K ) is a topological group underthe pointwise convergence topology.
DefinitionSuppose (G,K ) is a Hecke pair. The Schlichting completion of(G,K ) is defined to be G//K := σ(G).
Proposition (folklore)Let (G,K ) be a Hecke pair.
1 G//K is a t.d.l.c. group.2 If K is infinite with trivial normal core, then G//K is non-discrete.3 If G is finitely generated, then G//K is compactly generated.
Phillip Wesolek Commensurated subgroups 11/15/2016 3 / 18
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Motivation 1
When a group has few normal subgroups, commensurated subgroupsseem rather special.
1 SLn(Z) is commensurated in SLn(Z[1p ]).
2 Vaut is commensurated in V .
Theorem (Shalom–Willis, 13)
For n ≥ 3, every commensurated subgroup of SLn(Z[1p ]) is either finite,
commensurate with SLn(Z), or of finite index.
Proposition (Le Boudec–W., 16)Every proper commensurated subgroup of Thompson’s group T isfinite.
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Motivation 1
When a group has few normal subgroups, commensurated subgroupsseem rather special.
1 SLn(Z) is commensurated in SLn(Z[1p ]).
2 Vaut is commensurated in V .
Theorem (Shalom–Willis, 13)
For n ≥ 3, every commensurated subgroup of SLn(Z[1p ]) is either finite,
commensurate with SLn(Z), or of finite index.
Proposition (Le Boudec–W., 16)Every proper commensurated subgroup of Thompson’s group T isfinite.
Phillip Wesolek Commensurated subgroups 11/15/2016 4 / 18
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Motivation 1
When a group has few normal subgroups, commensurated subgroupsseem rather special.
1 SLn(Z) is commensurated in SLn(Z[1p ]).
2 Vaut is commensurated in V .
Theorem (Shalom–Willis, 13)
For n ≥ 3, every commensurated subgroup of SLn(Z[1p ]) is either finite,
commensurate with SLn(Z), or of finite index.
Proposition (Le Boudec–W., 16)Every proper commensurated subgroup of Thompson’s group T isfinite.
Phillip Wesolek Commensurated subgroups 11/15/2016 4 / 18
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Motivation 1
When a group has few normal subgroups, commensurated subgroupsseem rather special.
1 SLn(Z) is commensurated in SLn(Z[1p ]).
2 Vaut is commensurated in V .
Theorem (Shalom–Willis, 13)
For n ≥ 3, every commensurated subgroup of SLn(Z[1p ]) is either finite,
commensurate with SLn(Z), or of finite index.
Proposition (Le Boudec–W., 16)Every proper commensurated subgroup of Thompson’s group T isfinite.
Phillip Wesolek Commensurated subgroups 11/15/2016 4 / 18
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Motivation 1
When a group has few normal subgroups, commensurated subgroupsseem rather special.
1 SLn(Z) is commensurated in SLn(Z[1p ]).
2 Vaut is commensurated in V .
Theorem (Shalom–Willis, 13)
For n ≥ 3, every commensurated subgroup of SLn(Z[1p ]) is either finite,
commensurate with SLn(Z), or of finite index.
Proposition (Le Boudec–W., 16)Every proper commensurated subgroup of Thompson’s group T isfinite.
Phillip Wesolek Commensurated subgroups 11/15/2016 4 / 18
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Motivation 2
Via the Schlichting completion, commensurated subgroups can giveinteresting t.d.l.c. groups.
Proposition (Shalom–Willis, 13)
For n ≥ 2, SLn(Z[1p ])//SLn(Z) ' PSLn(Qp).
Proposition (folklore)V//Vaut = AAut(T2,2).
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Motivation 2
Via the Schlichting completion, commensurated subgroups can giveinteresting t.d.l.c. groups.
Proposition (Shalom–Willis, 13)
For n ≥ 2, SLn(Z[1p ])//SLn(Z) ' PSLn(Qp).
Proposition (folklore)V//Vaut = AAut(T2,2).
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Motivation 2
Via the Schlichting completion, commensurated subgroups can giveinteresting t.d.l.c. groups.
Proposition (Shalom–Willis, 13)
For n ≥ 2, SLn(Z[1p ])//SLn(Z) ' PSLn(Qp).
Proposition (folklore)V//Vaut = AAut(T2,2).
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Preliminaries
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Let α := (ai)i∈N be a sequence natural numbers with ai ≥ 2. Therooted tree Tα is defined to be the rooted tree such that a vertex onlevel n has an many children on level n + 1.• For G ≤ Aut(Tα) and s ∈ Tα, the rigid stabilizer of s is
ristG(s) := StabG({r ∈ Tα | r � s}).
• The n-th level rigid stabilizer of G is ristG(n) := 〈ristG(s) | |s| = n〉.
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Let α := (ai)i∈N be a sequence natural numbers with ai ≥ 2.
Therooted tree Tα is defined to be the rooted tree such that a vertex onlevel n has an many children on level n + 1.• For G ≤ Aut(Tα) and s ∈ Tα, the rigid stabilizer of s is
ristG(s) := StabG({r ∈ Tα | r � s}).
• The n-th level rigid stabilizer of G is ristG(n) := 〈ristG(s) | |s| = n〉.
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Let α := (ai)i∈N be a sequence natural numbers with ai ≥ 2. Therooted tree Tα is defined to be the rooted tree such that a vertex onlevel n has an many children on level n + 1.
• For G ≤ Aut(Tα) and s ∈ Tα, the rigid stabilizer of s is
ristG(s) := StabG({r ∈ Tα | r � s}).
• The n-th level rigid stabilizer of G is ristG(n) := 〈ristG(s) | |s| = n〉.
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Let α := (ai)i∈N be a sequence natural numbers with ai ≥ 2. Therooted tree Tα is defined to be the rooted tree such that a vertex onlevel n has an many children on level n + 1.• For G ≤ Aut(Tα) and s ∈ Tα, the rigid stabilizer of s is
ristG(s) := StabG({r ∈ Tα | r � s}).
• The n-th level rigid stabilizer of G is ristG(n) := 〈ristG(s) | |s| = n〉.
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Let α := (ai)i∈N be a sequence natural numbers with ai ≥ 2. Therooted tree Tα is defined to be the rooted tree such that a vertex onlevel n has an many children on level n + 1.• For G ≤ Aut(Tα) and s ∈ Tα, the rigid stabilizer of s is
ristG(s) := StabG({r ∈ Tα | r � s}).
• The n-th level rigid stabilizer of G is ristG(n) := 〈ristG(s) | |s| = n〉.
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).(ii) G acts transitively on each level of Tα.(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products, Grigorchuk group, Gupta-Sidki groups, . . . .
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).
(ii) G acts transitively on each level of Tα.(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products, Grigorchuk group, Gupta-Sidki groups, . . . .
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).(ii) G acts transitively on each level of Tα.
(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products, Grigorchuk group, Gupta-Sidki groups, . . . .
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).(ii) G acts transitively on each level of Tα.(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products, Grigorchuk group, Gupta-Sidki groups, . . . .
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).(ii) G acts transitively on each level of Tα.(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products,
Grigorchuk group, Gupta-Sidki groups, . . . .
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).(ii) G acts transitively on each level of Tα.(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products, Grigorchuk group,
Gupta-Sidki groups, . . . .
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DefinitionA group G is said to be a branch group if there is a rooted tree Tα sothat the following hold:
(i) G is isomorphic to a subgroup of Aut(Tα).(ii) G acts transitively on each level of Tα.(iii) For each level n, the index |G : ristG(n)| is finite.
ExamplesIterated wreath products, Grigorchuk group, Gupta-Sidki groups, . . . .
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Commensurated subgroups in branch groups
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An infinite group G is just infinite if every proper quotient is finite.
Theorem (W., 15)Let G be a finitely generated branch group. Then G is just infinite if andonly if every commensurated subgroup is either finite or of finite index.
CorollaryLet G be the Grigorchuk group or a Gupta-Sidki group. Everycommensurated subgroup of G is either finite or of finite index.
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An infinite group G is just infinite if every proper quotient is finite.
Theorem (W., 15)Let G be a finitely generated branch group. Then G is just infinite if andonly if every commensurated subgroup is either finite or of finite index.
CorollaryLet G be the Grigorchuk group or a Gupta-Sidki group. Everycommensurated subgroup of G is either finite or of finite index.
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An infinite group G is just infinite if every proper quotient is finite.
Theorem (W., 15)Let G be a finitely generated branch group. Then G is just infinite if andonly if every commensurated subgroup is either finite or of finite index.
CorollaryLet G be the Grigorchuk group or a Gupta-Sidki group. Everycommensurated subgroup of G is either finite or of finite index.
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Idea of proofThe reverse implication follows by work of Grigorchuk.
For the forwardimplication, we argue by contradiction.• Suppose K ≤ G is an infinite, infinite index commensurated
subgroup.• Consider the Schlichting completion G//K . This is a compactly
generated t.d.l.c. group that is non-compact and non-discrete.• Apply results for t.d.l.c. groups to derive a contradiction.
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Idea of proofThe reverse implication follows by work of Grigorchuk. For the forwardimplication, we argue by contradiction.
• Suppose K ≤ G is an infinite, infinite index commensuratedsubgroup.• Consider the Schlichting completion G//K . This is a compactly
generated t.d.l.c. group that is non-compact and non-discrete.• Apply results for t.d.l.c. groups to derive a contradiction.
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Idea of proofThe reverse implication follows by work of Grigorchuk. For the forwardimplication, we argue by contradiction.• Suppose K ≤ G is an infinite, infinite index commensurated
subgroup.
• Consider the Schlichting completion G//K . This is a compactlygenerated t.d.l.c. group that is non-compact and non-discrete.• Apply results for t.d.l.c. groups to derive a contradiction.
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Idea of proofThe reverse implication follows by work of Grigorchuk. For the forwardimplication, we argue by contradiction.• Suppose K ≤ G is an infinite, infinite index commensurated
subgroup.• Consider the Schlichting completion G//K .
This is a compactlygenerated t.d.l.c. group that is non-compact and non-discrete.• Apply results for t.d.l.c. groups to derive a contradiction.
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Idea of proofThe reverse implication follows by work of Grigorchuk. For the forwardimplication, we argue by contradiction.• Suppose K ≤ G is an infinite, infinite index commensurated
subgroup.• Consider the Schlichting completion G//K . This is a compactly
generated t.d.l.c. group that is non-compact and non-discrete.
• Apply results for t.d.l.c. groups to derive a contradiction.
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Idea of proofThe reverse implication follows by work of Grigorchuk. For the forwardimplication, we argue by contradiction.• Suppose K ≤ G is an infinite, infinite index commensurated
subgroup.• Consider the Schlichting completion G//K . This is a compactly
generated t.d.l.c. group that is non-compact and non-discrete.• Apply results for t.d.l.c. groups to derive a contradiction.
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Result 1
Theorem (Caprace–Monod, 10)For G a compactly generated t.d.l.c. group, one of the following holds:
1 G has an infinite discrete quotient.2 G is compact.3 G has a cocompact normal subgroup that admits exactly
0 < n <∞ non-discrete topologically simple quotients.
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Result 1
Theorem (Caprace–Monod, 10)For G a compactly generated t.d.l.c. group, one of the following holds:
1 G has an infinite discrete quotient.
2 G is compact.3 G has a cocompact normal subgroup that admits exactly
0 < n <∞ non-discrete topologically simple quotients.
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Result 1
Theorem (Caprace–Monod, 10)For G a compactly generated t.d.l.c. group, one of the following holds:
1 G has an infinite discrete quotient.2 G is compact.
3 G has a cocompact normal subgroup that admits exactly0 < n <∞ non-discrete topologically simple quotients.
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Result 1
Theorem (Caprace–Monod, 10)For G a compactly generated t.d.l.c. group, one of the following holds:
1 G has an infinite discrete quotient.2 G is compact.3 G has a cocompact normal subgroup that admits exactly
0 < n <∞ non-discrete topologically simple quotients.
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Result 2
For a closed normal factor K/L of a topological group G, thecentralizer is
CG(K/L) := {g ∈ G | [g,K ] ⊆ L}.
DefinitionFor a topological group G, closed normal factors K1/L1 and K2/L2 areassociated if CG(K1/L1) = CG(K2/L2).
DefinitionAn equivalence class of non-abelian chief factors under theassociation relation is called a chief block. The set of chief blocks isdenoted by BG.
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Result 2
For a closed normal factor K/L of a topological group G, thecentralizer is
CG(K/L) := {g ∈ G | [g,K ] ⊆ L}.
DefinitionFor a topological group G, closed normal factors K1/L1 and K2/L2 areassociated if CG(K1/L1) = CG(K2/L2).
DefinitionAn equivalence class of non-abelian chief factors under theassociation relation is called a chief block. The set of chief blocks isdenoted by BG.
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Result 2
For a closed normal factor K/L of a topological group G, thecentralizer is
CG(K/L) := {g ∈ G | [g,K ] ⊆ L}.
DefinitionFor a topological group G, closed normal factors K1/L1 and K2/L2 areassociated if CG(K1/L1) = CG(K2/L2).
DefinitionAn equivalence class of non-abelian chief factors under theassociation relation is called a chief block.
The set of chief blocks isdenoted by BG.
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Result 2
For a closed normal factor K/L of a topological group G, thecentralizer is
CG(K/L) := {g ∈ G | [g,K ] ⊆ L}.
DefinitionFor a topological group G, closed normal factors K1/L1 and K2/L2 areassociated if CG(K1/L1) = CG(K2/L2).
DefinitionAn equivalence class of non-abelian chief factors under theassociation relation is called a chief block. The set of chief blocks isdenoted by BG.
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A topological group G is Polish if the topology is separable and admitsa complete, compatible metric.
Theorem (Reid–W., 15)Let G be a Polish group, a ∈ BG, and
{1} = G0 ≤ G1 ≤ · · · ≤ Gn = G
be a series of closed normal subgroups in G. Then there is exactly onei ∈ {0, . . . ,n − 1} such that there exist closed normal subgroupsGi ≤ B < A ≤ Gi+1 of G for which A/B ∈ a.
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A topological group G is Polish if the topology is separable and admitsa complete, compatible metric.
Theorem (Reid–W., 15)Let G be a Polish group, a ∈ BG, and
{1} = G0 ≤ G1 ≤ · · · ≤ Gn = G
be a series of closed normal subgroups in G. Then
there is exactly onei ∈ {0, . . . ,n − 1} such that there exist closed normal subgroupsGi ≤ B < A ≤ Gi+1 of G for which A/B ∈ a.
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A topological group G is Polish if the topology is separable and admitsa complete, compatible metric.
Theorem (Reid–W., 15)Let G be a Polish group, a ∈ BG, and
{1} = G0 ≤ G1 ≤ · · · ≤ Gn = G
be a series of closed normal subgroups in G. Then there is exactly onei ∈ {0, . . . ,n − 1} such that there exist closed normal subgroupsGi ≤ B < A ≤ Gi+1 of G for which A/B ∈ a.
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A question
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A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
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A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
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A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
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A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.
3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
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A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.
4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
Phillip Wesolek Commensurated subgroups 11/15/2016 16 / 18
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A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.
5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
Phillip Wesolek Commensurated subgroups 11/15/2016 16 / 18
![Page 69: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/69.jpg)
A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
Phillip Wesolek Commensurated subgroups 11/15/2016 16 / 18
![Page 70: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/70.jpg)
A locally compact group is amenable if it admits a finitely additive leftinvariant Borel probability measure.
DefinitionLet A E be the smallest class of locally compact groups so that
1 A E contains all compact groups and all amenable discretegroups.
2 A E is closed under group extension.3 A E is closed under taking closed subgroups.4 A E is closed under taking Hausdorff quotients.5 A E is closed under directed unions of open subgroups.
QuestionIs every amenable locally compact group a member of A E ?
Phillip Wesolek Commensurated subgroups 11/15/2016 16 / 18
![Page 71: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/71.jpg)
One may be able to attack this question via the Schlichting completion.
ObservationSuppose (G,K ) is a Hecke pair. If G is amenable, then G//K isamenable.
ProblemFind finitely generated amenable groups with interestingcommensurated subgroups.
QuestionDoes the Basilica group have an infinite, infinite index commensuratedsubgroup with trivial normal core?
Phillip Wesolek Commensurated subgroups 11/15/2016 17 / 18
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One may be able to attack this question via the Schlichting completion.
ObservationSuppose (G,K ) is a Hecke pair. If G is amenable, then G//K isamenable.
ProblemFind finitely generated amenable groups with interestingcommensurated subgroups.
QuestionDoes the Basilica group have an infinite, infinite index commensuratedsubgroup with trivial normal core?
Phillip Wesolek Commensurated subgroups 11/15/2016 17 / 18
![Page 73: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/73.jpg)
One may be able to attack this question via the Schlichting completion.
ObservationSuppose (G,K ) is a Hecke pair. If G is amenable, then G//K isamenable.
ProblemFind finitely generated amenable groups with interestingcommensurated subgroups.
QuestionDoes the Basilica group have an infinite, infinite index commensuratedsubgroup with trivial normal core?
Phillip Wesolek Commensurated subgroups 11/15/2016 17 / 18
![Page 74: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/74.jpg)
One may be able to attack this question via the Schlichting completion.
ObservationSuppose (G,K ) is a Hecke pair. If G is amenable, then G//K isamenable.
ProblemFind finitely generated amenable groups with interestingcommensurated subgroups.
QuestionDoes the Basilica group have an infinite, infinite index commensuratedsubgroup with trivial normal core?
Phillip Wesolek Commensurated subgroups 11/15/2016 17 / 18
![Page 75: Commensurated subgroups of finitely generated branch groups · 2016. 11. 15. · Phillip Wesolek Commensurated subgroups 11/15/2016 2 / 18. Definition A subgroup K of a group G](https://reader036.vdocuments.mx/reader036/viewer/2022071412/6109507e8e049b27fe1dca9f/html5/thumbnails/75.jpg)
Thank you
Phillip Wesolek Commensurated subgroups 11/15/2016 18 / 18