176 research outputs found
Limits of (certain) CAT(0) groups, I: Compactification
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 R-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.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-52.abs.htm
Limit groups for relatively hyperbolic groups, I: The basic tools
We begin the investigation of Gamma-limit groups, where Gamma is a
torsion-free group which is hyperbolic relative to a collection of free abelian
subgroups. Using the results of Drutu and Sapir, we adapt the results from
math.GR/0404440 to this context. Specifically, given a finitely generated group
G, and a sequence of pairwise non-conjugate homomorphisms {h_n : G -> Gamma},
we extract an R-tree with a nontrivial isometric G-action.
We then prove an analogue of Sela's shortening argument.Comment: 41 pages. The new version of this paper has been substantially
rewritten. It now includes all of the results of the previous version, and
also of math.GR/0408080. The exception to this is the proof of the Hopf
property, which follows imediately from Theorem 5.2 of math.GR/0503045 (and
does not use anything omitted from this version
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