On the asymptotic geometry of abelian-by-cyclic groups

2000/05/17 by Benson Farb, Lee Mosher · 1 citation
Mathematics · #math.GR #msc:20F65 #msc:20F69

paper · pdf

published as Acta Mathematica 184 no. 2 (2000), 145--202 · 65 pages, 2 figures. To appear in Acta Mathematica

arxiv created 2000/05/17 · arxiv updated 2009/11/30

Abstract

A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Zⁿ, determined by an n x n integer matrix M with det(M) ≠ 0. The group Gamma_M is polycyclic if and only if |det(M)|=1. We give a complete classification of the nonpolycyclic groups Gamma_M up to quasi-isometry: given n x n integer matrices M,N with |det(M)|, |det(N)| > 1, the groups Gamma_M, Gamma_N are quasi-isometric if and only if there exist positive integers r,s such that M^r, N^s have the same absolute Jordan form. We also prove quasi-isometric rigidity: if Gamma_M is an abelian-by-cyclic group determined by an n x n integer matrix M with |det(M)| > 1, and if G is any finitely generated group quasi-isometric to Gamma_M, then there is a finite normal subgroup K of G such that G/K is abstractly commensurable to Gamma_N, for some n x n integer matrix N with |det(N)| > 1.

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• Rigidity and flexibility results for groups with a common cocompact envelope

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