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Limits of (certain) CAT(0) groups, I: Compactification

Daniel Groves

Algebraic & Geometric Topology 5 (2005) 1325–1364

arXiv: math.GR/0404440


The purpose of this paper is to investigate torsion-free groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats, as defined by Hruska. Our approach is to seek results analogous to those of Sela, Kharlampovich and Miasnikov for free groups and to those of Sela (and Rips and Sela) for torsion-free hyperbolic groups.

This paper is the first in a series. In this paper we extract an –tree from an asymptotic cone of certain CAT(0) spaces. This is analogous to a construction of Paulin, and allows a great deal of algebraic information to be inferred, most of which is left to future work.

CAT(0) spaces, isolated flats, limit groups, $\mathbb{R}$–trees
Mathematical Subject Classification 2000
Primary: 20F65
Secondary: 20F67, 20E08, 57M07
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Received: 14 January 2005
Accepted: 20 September 2005
Published: 6 October 2005
Daniel Groves
Department of Mathematics
California Institute of Technology
Pasadena CA 91125