[Paper Review] Codimension one subgroups and boundaries of hyperbolic groups

Thomas Delzant, Panos Papasoglu|ArXiv.org|Jul 18, 2008

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper constructs hyperbolic groups with arbitrarily high-dimensional boundaries that are separated by a Cantor set (dimension 0), yet do not split over any subgroup—demonstrating that Bowditch's theorem on splittings over 2-ended groups cannot be generalized to more complex subgroups. The construction uses small cancellation theory over free products, embedding torsion-free hyperbolic groups with property (T) and high boundary dimension into a new group via specific relators, ensuring codimension one subgroups exist but no nontrivial splittings occur.

ABSTRACT

We construct hyperbolic groups with the following properties: The boundary of the group has big dimension, it is separated by a Cantor set and the group does not split. This shows that Bowditch's theorem that characterizes splittings of hyperbolic groups over 2-ended groups in terms of the boundary can not be extended to splittings over more complicated subgroups.

Motivation & Objective

• To challenge the generalization of Bowditch's theorem on splittings over 2-ended groups to more complex subgroups.
• To construct hyperbolic groups whose boundaries have arbitrarily high topological dimension.
• To show that such groups can be separated by a Cantor set (dimension 0) yet not split over any subgroup.
• To demonstrate that codimension one subgroups do not imply group splittings in hyperbolic groups beyond the 2-ended case.
• To answer a question in [10] about coarse separation and asymptotic dimension in non-splitting groups.

Proposed method

• Use a torsion-free, 1-ended hyperbolic group A with property (T) and dim(∂A) ≥ n, such as a lattice in Sp(n,1).
• Take a second copy B of A and form the free product A*B.
• Apply small cancellation relations r_{i,j} = (a_i b_j)(a_i b_j^2)(a_i b_j^3)(a_i b_j^4) for all i,j to define the quotient group G.
• Prove that the resulting group G has a free codimension one subgroup H via a generalized version of Wise's small cancellation over free products.
• Use van-Kampen diagrams over free products and apply combinatorial lemmas (e.g., Greendlinger-type inequalities) to control area growth and prove linear isoperimetric inequality.
• Establish hyperbolicity of G by showing it satisfies a linear isoperimetric inequality using diagram area bounds and vertex degree estimates.

Experimental results

Research questions

• RQ1 Can hyperbolic groups have high-dimensional boundaries that are coarsely separated by a zero-dimensional set (Cantor set) without splitting?
• RQ2 Does the existence of a codimension one subgroup imply a nontrivial splitting in hyperbolic groups beyond the 2-ended case?
• RQ3 Can Bowditch’s boundary-based splitting theorem be extended to splittings over subgroups more complex than 2-ended groups?
• RQ4 Is it possible to construct a hyperbolic group with arbitrarily high boundary dimension that still has property FA (no nontrivial splitting)?
• RQ5 Can a group have a uniformly embedded subset of asymptotic dimension 1 that coarsely separates it, yet not split over any subgroup?

Key findings

• For any n > 0, there exists a one-ended hyperbolic group G with dim(∂G) ≥ n.
• The boundary ∂G is separated by a Cantor set (a zero-dimensional set), indicating a nontrivial topological separation.
• Despite the presence of codimension one subgroups and a separating Cantor set in the boundary, G has property FA and does not split over any subgroup.
• The group G is constructed as a small cancellation quotient of A*B, where A and B are torsion-free hyperbolic groups with property (T) and dim(∂A) ≥ n.
• The construction ensures that G satisfies a linear isoperimetric inequality, confirming its hyperbolicity.
• The example answers a question from [10] by showing a finitely presented group with asdim G > n, coarsely separated by a uniformly embedded set of asdim 1, and with no nontrivial splitting.