Research announcement: A combination theorem for acylindrical complexes of hyperbolic groups and Cannon-Thurston maps

TL;DR

This paper proves that fundamental groups of acylindrical complexes of hyperbolic groups are hyperbolic and establishes the existence of Cannon-Thurston maps for specific subcomplexes, advancing understanding of hyperbolic group structures.

Contribution

It introduces a combination theorem for acylindrical complexes of hyperbolic groups and demonstrates the existence of Cannon-Thurston maps in this context.

Findings

Fundamental groups of acylindrical complexes are hyperbolic.

Vertex groups are quasiconvex within these complexes.

Cannon-Thurston maps exist for certain subcomplexes.

Abstract

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is hyperbolic in which the vertex groups are quasiconvex. In the second part of the article, we prove the existence of Cannon-Thurston maps for certain subcomplexes of groups in acylindrical complexes of hyperbolic groups (see Theorem 0.4).