Complexes of Groups and Boundaries

TL;DR

This paper develops conditions to construct EZ-structures for fundamental groups of complexes of groups, enabling the creation of hyperbolic groups and describing their Gromov boundaries.

Contribution

It introduces a method to build EZ-structures from local groups in complexes of groups, facilitating the construction of hyperbolic groups with known boundaries.

Findings

Provided a procedure to obtain hyperbolic groups as fundamental groups of complexes of hyperbolic groups.

Described the Gromov boundary for these constructed hyperbolic groups.

Established conditions for building EZ-structures from local group structures.

Abstract

Given a complex of groups over a finite simplicial complex in the sense of Haefliger, we give conditions under which it is possible to build an EZ-structure in the sense of Farrell-Lafont for its fundamental group out of such structures for its local groups. As an application, we prove a combination theorem that yields a procedure for getting hyperbolic groups as fundamental groups of simple complexes of hyperbolic groups. The construction provides a description of the Gromov boundary of such groups.