40 research outputs found
Median structures on asymptotic cones and homomorphisms into mapping class groups
The main goal of this paper is a detailed study of asymptotic cones of the
mapping class groups. In particular, we prove that every asymptotic cone of a
mapping class group has a bi-Lipschitz equivariant embedding into a product of
real trees, sending limits of hierarchy paths onto geodesics, and with image a
median subspace. One of the applications is that a group with Kazhdan's
property (T) can have only finitely many pairwise non-conjugate homomorphisms
into a mapping class group. We also give a new proof of the rank conjecture of
Brock and Farb (previously proved by Behrstock and Minsky, and independently by
Hamenstaedt).Comment: final version, to appear in Proc. LM
Rank rigidity for CAT(0) cube complexes
We prove that any group acting essentially without a fixed point at infinity
on an irreducible finite-dimensional CAT(0) cube complex contains a rank one
isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube
complexes. We derive a number of other consequences for CAT(0) cube complexes,
including a purely geometric proof of the Tits Alternative, an existence result
for regular elements in (possibly non-uniform) lattices acting on cube
complexes, and a characterization of products of trees in terms of bounded
cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor
Groups acting on tree-graded spaces and splittings of relatively hyperbolic group
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic
cones of groups (when the cones have cut points). Since many questions about
endomorphisms and automorphisms of groups, solving equations over groups,
studying embeddings of a group into another group, etc. lead to actions of
groups on the asymptotic cones, it is natural to consider actions of groups on
tree-graded spaces. We develop a theory of such actions which generalizes the
well known theory of groups acting on R-trees. As applications of our theory,
we describe, in particular, relatively hyperbolic groups with infinite groups
of outer automorphisms, and co-Hopfian relatively hyperbolic groups
Groups acting on tree-graded spaces and splittings of relatively hyperbolic group
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups
Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank
We give another proof of the quasi-isometric classification theorem of non-uniform lattices in higher rank semisimple groups. We use the asymptotic cone and a class of flats (the logarithmic flats) which move away in the cusp with logarithmic speed