Coarse differentiation and quasi-isometries of a class of solvable Lie groups II

Abstract

In this paper, we continue with the results in [12] and compute the group of quasiisometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results.

Rigidity of Teichmüller space

“…The proof uses the ideas of coarse differentiation, previously developed in the context of Teichmüller space in [EMR13] which in turn is based on the work of Eskin-Fisher-Whyte [EFW12,EFW13]. Coarse differentiation was also previously used by Peng to study quasi-flats in solvable Lie groups in [Pen11a] and [Pen11b]. The important property of maximal rank is that near such a point of maximal rank, Teichmüller space is close to being isometric to a product of copies of H, with the supremum metric.…”

Day’s fixed point theorem, group cohomology and quasi-isometric rigidity

“…In this section we show how Theorem 3.3 can be used to prove Tukia-type Theorem 1.2 and Theorem 5.4 below. Additionally we show how Theorem 1.2 can be used to simplify some of the proofs of quasi-isometric rigidity found in [6] and [18,19]. Then we show how both Theorem 1.2 and Theorem 5.4 can be used to improve results on envelopes of abelian-by-cyclic groups found in [8].…”

Residual finiteness and strict distortion of cyclic subgroups of solvable groups

“…We have by the top of [8, page 1684] that if G is a cocompact lattice in R d ⋊ M R, then each element of G ∩ R d is strictly exponentially distorted, and since G ∩ R d ∼ = Z d , we are able to apply Theorem 1.1. More generally, Sol and the solvable Lie groups R d ⋊ M R are examples of what are known as nondegenerate, split abelian by abelian Lie groups for which similar statements can be made (see [16,17] for a precise definition). Hence, we have the following corollary.…”