Weak Width of Subgroups

Abstract

We say that the weak width of an infinite subgroup $H$ of $G$ in $G$ is $n$ if there exists a collection of $n$ strongly essentially distinct conjugates $\{H,g_1^{-1}H{g}_1,\cdots ,g_{n-1}^{-1}H{g}_{n-1}\}$ of $H$ in $G$ such that the intersection $H\cap g_i^{-1}H{g}_i$ is infinite for all $1\le i\le n-1$ and $n$ is maximal possible. We prove that a quasiconvex subgroup of a negatively curved group has finite weak width in the ambient group. We also give examples demonstrating that height, width, and weak width are different invariants of a subgroup.