Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant \subcorank X of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for each $t\in T$ the set $g^{-1}(t)$ has subexponential growth rate in $X$ and the topological dimension $\dim T=k$ is minimal among all such maps. Our main result is the inequality \hyprank X\le\subcorank X for a large class of metric spaces $X$ including all locally compact Hadamard spaces, where \hyprank X is maximal topological dimension of \di Y among all \CAT(-1) spaces $Y$ quasi-isometrically embedded into $X$ (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of \hyprank conjectured by M. Gromov, in particular, that any Riemannian symmetric space $X$ of noncompact type possesses no quasi-isometric embedding \hyp^n\to X of the standard hyperbolic space \hyp^n with n-1>\dim X-\rank X.Comment: 12 page

On the dimension growth of groups

Dimension growth functions of groups have been introduced by Gromov in 1999. We prove that every solvable finitely generated subgroups of the R. Thompson group $F$ has polynomial dimension growth while the group $F$ itself, and some solvable groups of class 3 have exponential dimension growth with exponential control. We describe connections between dimension growth, expansion properties of finite graphs and the Ramsey theory.Comment: 20 pages; v3: Erratum and addendum included as Section 9. We can only prove that the lower bound of the dimension growth of $F$ is exp sqrt(n). New open questions and comments are added. v4: The paper is completely revised. Dimension growth with control is introduced, connections with graph expansion and Ramsey theory are include

Invariant means for the wobbling group

Given a metric space $(X,d)$, the wobbling group of $X$ is the group of bijections $g:X\rightarrow X$ satisfying $\sup\limits_{x\in X} d(g(x),x)<\infty$. We study algebraic and analytic properties of $W(X)$ in relation with the metric space structure of $X$, such as amenability of the action of the lamplighter group $\bigoplus_{X} \mathbf Z/2\mathbf Z \rtimes W(X)$ on $\bigoplus_{X} \mathbf Z/2\mathbf Z$ and property (T).Comment: 8 pages. v3: final version, with new presentation; to appear in the Bulletin of the BM