# Triangles, squares and geodesics

## TL;DR

This paper investigates the extension of known properties of nonpositively curved triangle and square complexes to mixed complexes, aiming to unify theories of simplicial and cube complexes.

## Contribution

It explores how results on biautomatic fundamental groups extend to complexes built from both triangles and squares, bridging different nonpositive curvature theories.

## Key findings

- Results extend to mixed complexes with triangles and squares
- Progress towards unifying simplicial and cube complex theories
- Provides foundational steps for higher-dimensional generalizations

## Abstract

In the early 1990s Steve Gersten and Hamish Short proved that compact nonpositively curved triangle complexes have biautomatic fundamental groups and that compact nonpositively curved square complexes have biautomatic fundamental groups. In this article we report on the extent to which results such as these extend to nonpositively curved complexes built out a mixture of triangles and squares. Since both results by Gersten and Short have been generalized to higher dimensions, this can be viewed as a first step towards unifying Januszkiewicz and {\'S}wi{\.a}tkowski's theory of simplicial nonpositive curvature with the theory of nonpositively curved cube complexes.