Weak Width of Subgroups

TL;DR

This paper introduces the concept of weak width for subgroups, explores its properties in negatively curved groups, and distinguishes it from related invariants like height and width.

Contribution

It defines weak width for subgroups, proves its finiteness for quasiconvex subgroups in negatively curved groups, and compares it with other subgroup invariants.

Findings

Weak width is finite for quasiconvex subgroups in negatively curved groups.

Examples show that height, width, and weak width are distinct invariants.

The paper provides a new perspective on subgroup invariants in geometric group theory.

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 $1 \leq i \leq n-1$ 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.