Group Actions on CAT(0) Simplicial Complexes

TL;DR

This paper proves that groups acting on CAT(0) simplicial 3-complexes are biautomatic, expanding the class of known groups with this property and exploring the relationship between nonpositive curvature and algebraic properties.

Contribution

It establishes that groups acting on CAT(0) simplicial 3-complexes are biautomatic, providing new positive examples in the study of nonpositive curvature and group theory.

Findings

Groups acting on CAT(0) simplicial 3-complexes are biautomatic.

Supports the conjecture linking nonpositive curvature to biautomaticity.

Expands known classes of biautomatic groups.

Abstract

The notions of nonpositive curved spaces and biautomatic groups are generalizations of the geometric properties of hyperbolic spaces and computational properties of their fundamental groups. Given the mutual origins of these conditions, one might conjecture that groups acting on nonpositive curved spaces are biautomatic. While the conventional wisdom is that counterexamples should exist, some groups acting on nonpositively curved spaces have been shown to be biautomatic. This article adds to the list of positive examples by proving that groups acting on CAT(0) simplicial 3-complexes are biautomatic.