Widths of Subgroups
Rita Gitik and Mahan Mitra and Eliyahu Rips and Michah Sageev
Let $H$ be a subgroup of a group $G$. We say that the conjugates $$\{ g_i^{-1}Hg_i \mid1 \le i \le n\} $$ of $H$ are essentially distinct if $Hg_i \neq Hg_j $ for $i \neq j$. We say that the width of an infinite subgroup $H$ in $G$ is $n$ if there exists a collection of $n$ essentially distinct conjugates of $H$ such that the intersection of any two elements of the collection is infinite and $n$ is maximal possible. We define the width of a finite subgroup to be $0$. We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic $3$-manifolds satisfy the $k$-plane property for some $k$.