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Title page for ETD etd-04062005-201041


Type of Document Dissertation
Author Minasyan, Ashot
Author's Email Address minasyan@math.vanderbilt.edu
URN etd-04062005-201041
Title On Quasiconvex Subsets of Hyperbolic Groups
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Alexander Olshanskiy Committee Chair
Bruce Hughes Committee Member
Mark Sapir Committee Member
Mike Mihalik Committee Member
Tom Kephart Committee Member
Keywords
  • geometric group theory
Date of Defense 2005-04-01
Availability unrestricted
Abstract
A geodesic metric space $X$ is called hyperbolic if there exists $delta ge 0$ such that

every geodesic triangle $Delta$ in $X$ is $delta$-slim, i.e., each side of $Delta$ is contained

in a closed $delta$-neighborhood of the two other sides. Let $G$ be a group generated by a finite set $A$

and let $Gamma$ be the corresponding Cayley graph. The group $G$ is said to be word hyperbolic if $Gamma$ is a hyperbolic

metric space. A subset $Q$ of the group $G$ is called quasiconvex if for any geodesic $gamma$ connecting

two elements from $Q$ in $Gamma$, $gamma$ is contained in a closed $varepsilon$-neighborhood of $Q$ (for some fixed

$varepsilon ge 0$). Quasiconvex subgroups play an important role in the theory of hyperbolic groups and have been

studied quite thoroughly.

We investigate properties of quasiconvex subsets in word hyperbolic groups and generalize a number of results

previously known about quasiconvex subgroups. On the other hand, we establish and study a notion of quasiconvex subsets

that are small relatively to subgroups. This allows to prove a theorem concerning residualizing homomorphisms

preserving such subsets. As corollaries, we obtain several new embedding theorems for word hyperbolic groups.

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