 
Summary: TRIANGLES, SQUARES AND GEODESICS
RENA LEVITT AND JON MCCAMMOND
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 com
plexes 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 re
sults by Gersten and Short have been generalized to higher dimensions,
this can be viewed as a first step towards unifying Januszkiewicz and
´Swiatkowski's theory of simplicial nonpositive curvature with the theory
of nonpositively curved cube complexes.
1. Introduction
Many concepts in geometric group theory, including hyperbolic groups,
CAT(0) groups, and biautomatic groups, were developed to capture the geo
metric and computational properties of examples such as negatively curved
closed Riemannian manifolds and closed topological 3manifolds. While the
geometric and computational aspects of hyperbolic groups are closely in
terconnected, the relationship between the geometrically defined class of
CAT(0) groups and the computationally defined class of biautomatic groups
