A finitely-presented solvable group with a small quasi-isometry group

We exhibit a family of infinite, finitely-presented, nilpotent-byabelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic Lie group. In addition, we propose a candidate for a polycyclic group whose quasi-isometry group is a solvable real Lie group, and we introduce a candidate for a quasi-isometrically rigid solvable group that is not finitely presented. We also record some conjectures on the large-scale geometry of lamplighter groups. Let Bn be the upper-triangular subgroup of SLn, and let Bn ≤ PGLn be the image of Bn under the natural quotient map SLn → PGLn. If p is a prime number, then the group Bn(Z[1/p]) is finitely presented for all n ≥ 2. In particular it is finitely generated, so we can form its quasiisometry group—denoted QI ( Bn(Z[1/p]) ) . In this paper we will prove Theorem A. If n ≥ 3 then QI ( Bn(Z[1/p]) ) ∼= ( Bn(R)×Bn(Qp) ) ⋊ Z/2Z The Z/2Z-action defining the semi-direct product above is given by a Qautomorphism of PGLn which stabilizes Bn. The order 2 automorphism acts simultaneously on each factor. ∗Supported in part by an N.S.F. Postdoctoral Fellowship.