On the asymptotic geometry of abelian-by-cyclic groups

Details On the asymptotic geometry of abelian-by-cyclic groups On the asymptotic geometry of abelian-by-cyclic groups

2000-05-17

arxiv.org/abs/math/0005181v1

Acta Mathematica 184 no. 2 (2000), 145--202

Mathematics

Group Theory

65 pages, 2 figures. To appear in Acta Mathematica

Scientific paper

A finitely presented, torsion free, abelian-by-cyclic group can always be written as an ascending HNN extension Gamma_M of Z^n, determined by an n x n integer matrix M with det(M) \ne 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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