A rigidity theorem for the solvable Baumslag-Solitar groups

Quasi-isometries of groups, graphs and surfaces

“…[18,11]] Let N be a set, let (X 1 , d 1 ) and (X 2 , d 2 ) be geodesic metric spaces with one-to-one maps f i : N → X i such that Proof. We need only show that the identity map between (N, d 1 ) and (N, d 2 ) is a quasi-isometry.…”

“…The answer is known in many cases, e.g. when X is a symmetric space (see for instance the survey [8]) or for some spaces arising in 3-manifold geometry [21,22,24] or combinatorial group theory [11,9,10,27]. Here we address a related question: given a smooth manifold R , determine the class of groups that are quasi-isometric to some complete Riemannian metric on R .…”

The large scale geometry of the higher Baumslag-Solitar groups

“…Thus the only graph of Zs with unbounded height function not coarsely orientation preserving quasi-isometric to the oriented tree of type (2, 2) is a graph of Zs with a single vertex and a single edge which includes isomorphically at one end. These are precisely the solvable Baumslag-Solitar group, which are classified up to quasi-isometry in [FM1].…”

Abstract Commensurators of Solvable Baumslag–Solitar Groups

“…The Baumslag -Solitar groups BS(m, n), 1 m n, are given by the presentation a, b | a −1 b m a = b n . These groups have served as a proving ground for many new ideas in combinatorial and geometric group theory (see, for instance, [2,5,6]). The only solvable groups in this class are groups BS(1, n); the groups BS(m, n) with 1 < m n contain a free nonabelian group.…”

“…In [5], Farb and Mosher proved for n 2 that QI(BS(1, n)) ∼ = Bilip(R) × Bilip(Q n ), where Q n is the metric space of n-adic rationals with the usual metric and Bilip(Y ) denotes the group of bilipschitz homeomorphisms of a metric space Y . Moreover, they proved that BS(1, n) and BS(1, k) with n, k 1 are quasi-isometric if and only if these groups are commensurable, that happens if and only if n and k have common powers.…”