Hyperbolic groups and their subgroups
A group is of type if it admits a classifying space with finite -skeleton. For instance, iff finitely generated. iff finitely presented.
Question (Brady). Do there exist groups of type and not with no subgroups ?
Today, I will focus on looking for examples as subgroups of hyperbolic groups.
The answer in case follows from this classical thm:
Theorem 1 (Rips) Given a finitely presented group , there exists a hyperbolic group and a surjective map with finitely generated kernel.
Furthermore, if is infinite, is not finitely presented.
Theorem 2 (Brady 1999, Lodha 2017) There exist infinitely many pairwise non isomorphic hyperbolic groups, each containing a subgroup of type , not .
No result for . The following result indicates that it is harder.
Theorem 3 (Gersten) If has cohomological dimension 2, is hyperbolic, any finitely presented subgroup of is hyperbolic.
Definition 4 A graph is sizable if
- is bipartite on and , , .
- has no 4-cycles.
- is connected for all .
Such graphs exist, a computer search has produced one with 31 vertices.
Theorem 5 Given sizable graphs , , , there exists a cube complex such that is hyperbolic, maps onto with not kernel.
One needs hyperbolic groups 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 is systolic, a finitely presented subgroup of , then is systolic (and thus for all ).
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 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 -manifold, , is 5-large). Therefore, I want to relax 5-large.
Definition 8 A pair of -partite (i.e. contained in the join of graphs) flag complexes are pairwise 5-large if
- Every 4-cycle in (or ) is contained in a 2-colored part of (resp. ).
- For each pair , either or has no 4-cycle.
Pairwise 5-large allows to construct with hyperbolic fundamental group. Indeed, the -bipartite structure un and allows to get an -embedding a product of trees. A flat in , when mapped to one tree factor, falls onto a line, whence a map to . We show this is an -embedding, the image is a plane which cannot be transverse to coordinate hyperplanes, contradiction. Therefore is hyperbolic.