A gluing theorem for negatively curved complexes

TL;DR

This paper proves a gluing theorem for negatively curved complexes, showing that under certain conditions, complex constructions from negatively curved 2-complexes preserve negative curvature, impacting the understanding of hyperbolic groups.

Contribution

It introduces a new gluing theorem for negatively curved complexes, extending the class of groups known to be CAT(-1) and providing tools for constructing hyperbolic groups.

Findings

Certain groups are proven to be CAT(-1)

Gluing negatively curved complexes preserves negative curvature

Applications to hyperbolic limit groups and JSJ decompositions

Abstract

A simplicial complex is called negatively curved if all its simplices are isometric to simplices in hyperbolic space, and it satisfies Gromov's Link Condition. We prove that, subject to certain conditions, a compact graph of spaces whose vertex spaces are negatively curved 2-complexes, and whose edge spaces are points or circles, is negatively curved. As a consequence, we deduce that certain groups are CAT(-1). These include hyperbolic limit groups, and hyperbolic groups whose JSJ components are fundamental groups of negatively curved 2-complexes---for example, finite graphs of free groups with cyclic edge groups.