Notes of Robert Kropholler’s Cambridge lecture 10-01-2017

Hyperbolic groups and their subgroups

A group is of type {F^n} if it admits a classifying space with finite {n}-skeleton. For instance, {F^1} iff finitely generated. {F^2} iff finitely presented.

Question (Brady). Do there exist groups of type {F^n} and not {F^{n+1}} with no {{\mathbb Z}^2} subgroups ?

Today, I will focus on looking for examples as subgroups of hyperbolic groups.

1. Results

1.1. {n=1}

The answer in case {n=1} follows from this classical thm:

Theorem 1 (Rips) Given a finitely presented group {Q}, there exists a hyperbolic group {G} and a surjective map {G\rightarrow Q} with finitely generated kernel.

Furthermore, if {Q} is infinite, {N} is not finitely presented.

1.2. {n=2}

Theorem 2 (Brady 1999, Lodha 2017) There exist infinitely many pairwise non isomorphic hyperbolic groups, each containing a subgroup of type {F^2}, not {F^3}.

No result for {n\geq 3}. The following result indicates that it is harder.

Theorem 3 (Gersten) If {G} has cohomological dimension 2, is hyperbolic, any finitely presented subgroup of {G} is hyperbolic.

1.3. Construction

Definition 4 A graph {\Gamma} is sizable if

  1. {\Gamma} is bipartite on {A} and {B}, {A=A^+\coprod A^-}, {B=B^+\coprod B^-}.
  2. {\Gamma} has no 4-cycles.
  3. {\Gamma(A^s \cup B^t)} is connected for all {s,t=\pm}.

Such graphs exist, a computer search has produced one with 31 vertices.

Theorem 5 Given sizable graphs {\Gamma_i}, {i=1,2,3}, {\Gamma_i\subset A_i *B_i}, there exists a {CAT(0)} cube complex {X\subset \prod_{i=1}^3 A_i *B_{i+1}} such that {G=\pi_1(X)} is hyperbolic, {G} maps onto {{\mathbb Z}} with {F^r} not {F^s} kernel.

1.4. Observation

One needs hyperbolic groups {G} with high geometric dimension. Here are examples: Januskiewicz-Swiatkowski 2003, 2006. Haglund 2003. Arzhantseva-Bridson-Januskiewicz-Leary-Minasyan-Swiatkowski 2009. All these examples produce systolic groups, they cannot be used, because of following

Theorem 6 (Wise, Zadnik) If {G} is systolic, {N} a finitely presented subgroup of {G}, then {N} is systolic (and thus {F^n} for all {n}).

Instead, we shall rely on the following family of examples.

Theorem 7 (Osajda 2010) There exist right angled Coxeter groups of arbitrary virtual cohomological dimension.

Such a group {W_L} is hyperbolic iff the graph is 5-large (i.e. triangle and square-free).

5-large graphs are scarce (for instance, no triangulation of an {n}-manifold, {n\geq 5}, is 5-large). Therefore, I want to relax 5-large.

Definition 8 A pair of {n}-partite (i.e. contained in the join of {n} graphs) flag complexes {A,B} are pairwise 5-large if

  1. Every 4-cycle in {A} (or {B}) is contained in a 2-colored part of {A} (resp. {B}).
  2. For each pair {(i,j)}, either {A(A_i\cup A_j)} or {B(B_i\cup B_j)} has no 4-cycle.

Pairwise 5-large allows to construct {X\subset \prod_{i=1}^3 A_i *B_{i}} with hyperbolic fundamental group. Indeed, the {n}-bipartite structure un {A} and {B} allows to get an {L^1}-embedding {\tilde X\rightarrow} a product of {n} trees. A flat in {\tilde X}, when mapped to one tree factor, falls onto a line, whence a map to {{\mathbb R}^n}. We show this is an {L^2}-embedding, the image is a plane which cannot be transverse to coordinate hyperplanes, contradiction. Therefore {\tilde X} is hyperbolic.

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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/
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