On the biautomaticity of CAT(0) triangle-square groups

TL;DR

This paper explores the biautomaticity of groups acting on CAT(0) triangle-square complexes, providing counterexamples to previous conjectures and establishing biautomaticity for one case while leaving another open.

Contribution

It presents counterexamples to existing conjectures on geodesic methods for proving biautomaticity and proves biautomaticity for one specific complex's fundamental group.

Findings

Counterexamples to Gersten-Short geodesics approach

Proved biautomaticity of π₁(X₁)

Biautomaticity of π₁(X₂) remains open

Abstract

Following the research from the paper "Triangles, squares and geodesics" (arXiv:0910.5688) of Rena Levitt and Jon McCammond we investigate the properties of groups acting on CAT(0) triangle-square complexes, focusing mostly on biautomaticity of such groups. In particular we show two examples of nonpositively curved triangle-square complexes $X_1$ and $X_2$, such that their universal covers violate conjectures given in the aforementioned paper. This shows that the Gersten-Short geodesics cannot be used as a way of proving biautomaticity of groups acting on such complexes. Lastly we give a proof of biautomaticity of $\pi_1(X_1)$, however the biautomaticity of $\pi_1(X_2)$ remains unknown.