Some groups having only elementary actions on metric spaces
with hyperbolic boundaries
(with G. Noskov)
Geom. Dedicata, 104 (2004) 119-137.
We study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SL n (Zopf), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SL n (Zopf) for nge 3 cannot act on a visibility space X without fixing a point in $$\overline X $$ . Corollaries concern Floyd''s group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included.
(Another consequence is that these groups are not relatively hyperbolicity; seealso a preprint by Behrstock, Drutu, Mosher.)
The Bass conjecture and growth in groups
(with M. Neuhauser)
Colloq. Math. 100 (2004) 23-27.
We discuss Bass's conjecture on the vanishing of the Hattori-Stallings rank
from the viewpoint of geometric group theory. It is noted that groups without
u-elements satisfy this conjecture. This leads in particular to a simple proof
of the conjecture for groups of subexponential growth.
Free subgroups of groups with nontrivial Floyd boundary
Comm. Algebra, 31 (2003) 5361-5376.
We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205-218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.
See also Group actions.