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 AbstractA geodesic metric space $X$ is called hyperbolic if there exists $delta ge 0$ such thatevery 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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