Notes of Pierre-Emmanuel Caprace’s fourth Cambridge lecture 27-04-2017

Exotic lattices and simple locally compact groups, IV

1. Subgroup separability and residual finiteness

Theorem 1 (Kropholler-Reid-Wesolek-Caprace) Let {G} be a finitely generated group, let {H<G} be a commensurated subgroup (every conjugate commensurates {H}). Let {\tilde H} be the profinite closure of {H}, i.e. the intersection of all finite index subgroups of {G} containing {H}. Then the intersection of all conjugates of {\tilde H} has finite index in {\tilde H}.

1.1. Proof

{\tilde H} is commensurated as well. Therefore one can assume that {H} is closed. For {X\subset G}, let {H_X} be the intersection of conjugates of {H} by elements of {X}. We must show that {H_G} has finite index in {H}.

Since {H} is commensurated, for every finite set {X} containing 1, {H_X} has finite index in {H}. Since {H} is closed, each {H_X} is separable, hence {H_X=W\cap H} where {W} is a finite index in {G}. Let {V} be a finite index normal subgroup of {G} contained in {W}. Then {H_X\subset VH_X\cap H\subset W\cap H=H_X}, hence {VH_X\cap H=H_X}.

We apply this to a fixed generating set containing 1, and get a normal subgroup {V}. Let {\mathcal{Y}} be the set of finite subsets containing 1 such that {VH_Y\cap H=H_Y}. Then {\mathcal{Y}} is stable under finite unions. Similarly, if {Y\in\mathcal{Y}} and {\{1,t\}\in\mathcal{Y}}, then {Y\cup tY\in \mathcal{Y}}. These two properties imply that all balls {X^n} belong to {\mathcal{Y}}. Therefore {H\cap V\subset VH_{X^n}\cap H=H_{X^n}}, hence {H\cap V\subset H_G\subset H}, {H_G} contains a finite index subgroup of {H}.

Our inital proof with Monod used more structure of locally compact groups.

1.2. Application to lattices in products of trees

Proposition 2 Let {T_i} be two leafless trees, let {\Gamma} act discretely cocompactly on their product. Then the following are equivalent.

  1. {\Gamma} is reducible.
  2. The projection of {\Gamma} on a factor is discrete.
  3. For every vertex {v} of one of the trees, the stabilizer {\Gamma_v} is separable in {\Gamma}.
  4. There exists a vertex {v} of one of the trees whose stabilizer {\Gamma_v} is separable in {\Gamma}.

The implication (2){\Rightarrow}(1) uses Burger-Mozes’ result that for all normal subgroups {N} of {G}, {\Gamma N/N} is a lattice in {G/N} iff {\Gamma\cap N} is a lattice in {N}.

The implication (4){\Rightarrow}(2) uses Theorem 1.

Corollary 3 Let {G} be a finitely generated group such that every infinite normal subgroup of {G} has trivial centralizer. If {G} is residually finite, then every commensurated subgroup {H} of {G} has trivial quasi-centralizer (i.e. no element commutes with a finite index subgroup of {H}).

This gives a fast proof that Baumslag-Solitar group {BS(m,n)} is residually finite iff {|m|=1} or {|n|=1} or {|m|=|n|} (Meskin). Indeed, this group acts faithfully on its Bass-Serre tree with no fixed points at infinity. Therefore normal subgroups have trivial centralizers. The cyclic group generated by {a} is commensurated and quasicentralizes itself, thus the ambient group cannot be residually finite.

Corollary 4 Let {G} be a finitely generated group such that every infinite normal subgroup of {G} has trivial centralizer. Let {H<G} be infinite and commensurated. If {G} is residually finite, then every normal subgroup of {G} cuts an infinite subgroup of {H}.

This gives a proof of the following result of Burger and Mozes.

Proposition 5 (Burger-Mozes) Let {\Gamma} be discrete and cocompact on product of two trees. If {\Gamma} is irreducible and residually finite, then the projection to each factor is injective.

Recall the group {\Gamma_2=\langle a,b,x,y\;|\;\ldots\rangle} I introduced in the first lecture. Then {x^3} and {y^3} both commute with an index 2 subgroup of {\langle a,b \rangle}. Therefore they act trivially on the vertical tree.

Next time, I will explain the proof that {\Gamma_2} is irreducible. We will then be able to conclude that {\Gamma_2} is not residually finite.

Advertisements

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
This entry was posted in Course and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s