Superrigidity, generalized harmonic maps, and
uniformly convex spaces
(with T. Gelander and G.A. Margulis)
GAFA 17 (2008) 1524-1550
We prove several superrigidity results for isometric actions on
Busemann non-positively curved uniformly convex metric spaces. In particular we generalize
some recent theorems of N. Monod on uniform and certain
non-uniform irreducible lattices in products of locally compact
groups, and we give a proof of an unpublished result on
commensurability superrigidity due to G.A. Margulis. The proofs rely on
certain notions of harmonic maps and the study of their existence,
uniqueness, and continuity.
On the dynamics of isometries
Geometry & Topology, 9 (2005) 2359-2394.
We provide an analysis of the dynamics of isometries and semicontractions of
metric spaces. Certain subsets of the boundary at infinity play a fundamental
role and are identified completely for the standard boundaries of CAT(0)
spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex
domains, and partially for Thurston’s boundary of Teichmuller spaces. We
present several rather general results concerning groups of isometries, as well
as the proof of other more specific new theorems, for example concerning the
existence of free nonabelian subgroups in CAT(0) geometry, iteration of holomorphic
maps, a metric Furstenberg lemma, random walks on groups, noncompactness
of automorphism groups of convex cones, and boundary behaviour of
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.)
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.
Boundaries and random walks on finitely generated infinite groups
Matematik, 41 (2003) 295-306.
We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ?? and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.