A dynamical approach to von Neumann dimension

Let G be an amenable group and V be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of l^2(G;V) (with respect to G) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces Y of l^p(G;V) a real positive number dim_{l^p} Y (which is the von Neumann dimension when p=2). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective G-equivariant linear map of finite-type from l^p(G;V) -> l^p(G; V') if dim V > dim V'. A generalization of the Ornstein-Weiss lemma is developed along the way.Comment: 23 pages. Mistake corrected in statement of P

A dynamical approach to von Neumann dimension

International audienceLet $\Gamma$ be an amenable group and $V$ be a finite dimensional vector space. Gromov pointed out that the von Neumann dimension of linear subspaces of $\ell^2(\Gamma;V)$ (with respect to $\Gamma$) can be obtained by looking at a growth factor for a dynamical (pseudo-)distance. This dynamical point of view (reminiscent of metric entropy) does not requires a Hilbertian structure. It is used in this article to associate to a $\Gamma$-invariant linear subspaces $Y$ of $\ell^p(\Gamma;V)$ a real positive number \dlp Y (which is the von Neumann dimension when $p=2$). By analogy with von Neumann dimension, the properties of this quantity are explored to conclude that there can be no injective $\Gamma$-equivariant linear map of finite-type from $\ell^p(\Gamma;V) \to \ell^p(\Gamma; V')$ if $\dim V > \dim V'$. A generalization of the Ornstein-Weiss lemma is developed along the way

A remark on the connectedness of spheres in Cayley graphs

The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let $B_n^{c,\infty}$ be the infinite connected component of the complement of $B_n$. Then G has connected spheres if there exists a $r >0$ such that $B_{n+r} \cap B_n^{c,\infty}$ is connected for all $n \geq 0$. This note shows that if G is finitely presented then it has connected spheres.Comment: 5p., 1 figur

The Liouville property and Hilbertian compression

Lower bound on the equivariant Hilbertian compression exponent $\alpha$ are obtained using random walks. More precisely, if the probability of return of the simple random walk is $\succeq \textrm{exp}(-n^\gamma)$ in a Cayley graph then $\alpha \geq (1-\gamma)/(1+\gamma)$. This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if $|B_n| \preceq e^{n^\nu}$ then the speed is $\preceq n^{1/(2-\nu)}$. Under a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to $\alpha \geq 1-\gamma$. Using a result from Naor and Peres on compression and the speed of random walks, this yields very promising bounds on speed and implies the Liouville property if $\gamma <1/2$.Comment: 16 page