 
Summary: LIMIT GROUPS ARE CAT(0)
EMINA ALIBEGOVI
C AND MLADEN BESTVINA
Abstract. We prove that every limit group acts geometrically on
a CAT(0) space with the isolated
ats property.
1. Introduction
A group is said to be a CAT(0) group if it acts geometrically, i.e.
properly discontinuously and cocompactly by isometries, on a CAT(0)
space. One should think of a CAT(0) space as a geodesic metric space
in which every geodesic triangle is at least as thin as its comparison
triangle in Euclidean plane. For basic facts about CAT(0) spaces and
groups a general reference is [BH99]. We will be interested mainly in
geodesic spaces which are locally CAT(0) spaces (nonpositively curved
spaces), i.e. every point has a neighborhood which is a CAT(0) space.
All of our (locally) CAT(0) spaces will be proper. In this paper we
show that the class of groups known as limit groups is CAT(0), thus
answering a question circulated by Z. Sela.
Limit groups arise naturally in the study of equations over free
groups. Historically they appear under dierent names: 9free groups
[Rem89], fully residually free groups [KM98a], [KM98b], !residually
