Quasi - Isometric Rigidity for Psl

1. Introduction. Combining the work of many people yields a complete quasi-isometry classification of irreducible lattices in semisimple Lie groups (see [F] for an overview of these results). One of the first general results in this classification is the complete description, up to quasi-isometry, of all nonuniform lattices in semisimple Lie groups of rank 1, proved by R. Schwartz [S1]. He shows that every quasi-isometry of such a lattice is equivalent to a unique commensurator of. (A commensurator of ⊂ G is an element g ∈ G so that ggg −1 ∩ has finite index in .) We call this result commensurator rigidity, although it is a different notion than the commensurator rigidity of Margulis. In [FS] it was conjectured that commensurator rigidity, or at least a slightly weaker statement, " quasi-isometric if and only if commensurable, " should apply to nonuniform lattices in a wide class of Lie groups. Here we prove that both of these statements are true for PSL 2 (Z[1/p]). In a different direction, B. Farb and L. Mosher proved analogous quasi-isometric rigidity results for the solvable Baumslag-Solitar groups. These groups are given by the presentation BS(1, n) = =a, b | aba −1 = b n and are not lattices in any Lie group. The group PSL 2 (Z[1/p]) is a nonuniform (i.e., noncocompact) lattice in the group PSL 2 (R)×PSL 2 (Q p), analogous to the classical Hilbert modular group PSL 2 (ᏻ d) in PSL 2 (R)×PSL 2 (R). It is also a basic example of an S-arithmetic group. The proofs of Theorems A, B, and C (stated below) combine techniques from the two types of quasi-isometric rigidity results mentioned above. When we construct a space p on which PSL 2 (Z[1/p]) acts properly, discontinuously, and cocompactly by isometries, we see that the horospheres forming the boundary components of p carry the geometry of the group BS(1, p). In this way the results of [FM] play a role in the quasi-isometric rigidity of PSL 2 (Z[1/p]). 1.1. Statement of results. In this paper we prove the following quasi-isometric rigidity results for the finitely generated groups PSL 2 (Z[1/p]), where p is a prime. Theorem A may be viewed as a strengthening of strong (Mostow) rigidity for PSL 2 (Z[1/p]). [M] Theorem A (Main theorem). Every quasi-isometry of PSL 2 (Z[1/p]) is equivalent to a commensurator of PSL 2 (Z[1/p]). Hence the natural map …