Atlas: Quasi-isometry classification of PSL(2,Z[1/p]) and PSL(2,Z[1/p,1/q]) by Jennifer Taback

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1999 Spring Topology Conference
March 18-20, 1999
University of Utah
Salt Lake City, UT, USA

Organizers
Mladen Bestvina, Greg Conner, Misha Kapovich, Bruce Kleiner

Quasi-isometry classification of PSL(2, Z[1/p]) and PSL(2, Z[1/p, 1/q])
by
Jennifer Taback
University of California-Berkeley

Quasi-isometry classification of PSL(2, Z[1/p]) and PSL(2, Z[1/p, 1/q])

I will describe the geometry of the groups PSL(2, Z[1/p]) and PSL(2, Z[1/p, 1/q]). Each geometric model of PSL(2, Z[1/p]) and PSL(2, Z[1/p, 1/q]) has a boundary consisting of ``horospheres''. For PSL(2, Z[1/p]) these horospheres exhibit the geometry of the solvable Baumslag-Solitar group BS(1, p²) = < a, b|aba^-1 = b^p² > . For PSL(2, Z[1/p, 1/q]) these horospheres exhibit the geometry of the group

G = < a, b, c | aba^-1 = b^p², cbc^-1 = b^q², ac=ca > .

I will prove that PSL(2, Z[1/p]) and PSL(2, Z[1/q]) are quasi-isometric iff p=q and discuss the generalization of this theorem to PSL(2, Z[1/p, 1/q]). The proofs will rely on the geometric models of these groups.

Date received: March 9, 1999