Xiangdong Xie
Bowling Green State University
Title
Pattern rigidity in nilpotent groups
Abstract
A pattern in a group consists of the left cosets of a collection of subgroups. A quasi-isometry between two groups with patterns is pattern-preserving if there is a constant D such that the image of every left coset in the domain pattern is at Hausdorff distance at most D from a left closet in the target pattern. There are two natural questions concerning PPQIs(pattern-preserving quasi-isometries). The first is to determine when there is a PPQI between two groups with patterns. The second is whether PPQIs exhibit rigidity properties. We will discuss the second question in the setting of nilpotent groups. We show that every self PPQI is at finite distance from an automorphism if the subgroups intersect the center trivially and generate the whole group and one of the following holds: (1) G is a 2-step torsion free finitely generated nilpotent group; (2) G is a 2-step simply connected nilpotent Lie group.
This is ongoing joint work with Mitra Alizadeh, Hao Liang and Qingshan Zhou.