On the asymptotic geometry of abelianbycyclic groups
Abstract
A finitely presented, torsion free, abelianbycyclic 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 quasiisometry: given n x n integer matrices M,N with det(M), det(N) > 1, the groups Gamma_M, Gamma_N are quasiisometric 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 quasiisometric rigidity: if Gamma_M is an abelianbycyclic group determined by an n x n integer matrix M with det(M) > 1, and if G is any finitely generated group quasiisometric 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2000
 arXiv:
 arXiv:math/0005181
 Bibcode:
 2000math......5181F
 Keywords:

 Group Theory;
 20F65 (primary);
 20F69 (secondary)
 EPrint:
 65 pages, 2 figures. To appear in Acta Mathematica