1,436 research outputs found
On the asymptotic geometry of abelian-by-cyclic groups
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.Comment: 65 pages, 2 figures. To appear in Acta Mathematic
Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II
Let BS(1,n)= . We prove that any finitely-generated
group quasi-isometric to BS(1,n) is (up to finite groups) isomorphic to
BS(1,n). We also show that any uniform group of quasisimilarities of the real
line is bilipschitz conjugate to an affine group.Comment: 42 pages. see also http://www.math.uchicago.edu/~far
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