 
Summary: CONSTRUCTING NONPOSITIVELY
CURVED SPACES AND GROUPS
JON MCCAMMOND 1
Abstract. The theory of nonpositively curved spaces and groups is tremen
dously powerful and has enormous consequences for the groups and spaces
involved. Nevertheless, our ability to construct examples to which the theory
can be applied has been severely limited by an inability to test in real time
whether a random finite piecewise Euclidean complex is nonpositively curved.
In this article I focus on the question of how to construct examples of non
positively curved spaces and groups, highlighting in particular the boundary
between what is currently doable and what is not yet feasible. Since this is in
tended primarily as a survey, the key ideas are merely sketched with references
pointing the interested reader to the original articles.
Over the past decade or so, the consequences of nonpositive curvature for geo
metric group theorists have been throughly investigated, most prominently in the
book by Bridson and Haefliger [26]. See also the recent review article by Kleiner
in the Bulletin of the AMS [59] and the related books by Ballmann [4], Ballmann
GromovSchroeder [5] and the original long article by Gromov [48]. In this article
I focus not on the consequences of the theory, but rather on the question of how
to construct examples to which it applies. The structure of the article roughly
