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Reduced cohomology motivates to look at harmonic functions which satisfy certain gradient conditions. If $G$ is a direct product of two infinite groups or a (FC-central)-by-cyclic group, then there are no harmonic functions with gradient in $c_0$ on its Cayley graphs. From this, it follows that a metabelian group $G$ has no harmonic functions with gradient in $\ell^p$. Furthermore, under a radial isoperimetric condition, groups whose isoperimetric profile is bounded by power of logarithms also have no harmonic functions with gradient in $\ell^p$., Comment: 39 pages, updated references

Balancing square and rectangular tables by rotation has been a interesting way to illustrate the intermediate value theorem. The aim of this note is to show that the balancing act but with non-rectangular tables can be a nice application of the ergodic theorem (or more generally, invariant measures)., Comment: 5 pages

We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Tim\'ar) for various graphs using the isoperimetric profile, growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile $\mathrm{Sep}(n)$ is also bounded by powers of $n$. For many amenable groups, we show a lower bound in $n/ \log(n)^a$ and, for any group which has a non-trivial compression exponent in an $L^p$-space, an upper bound in $n/ \log(n)^b$. We show that solvable groups of exponential growth cannot have a separation profile bounded above by a sublinear power function. In an appendix, we introduce the notion of local separation, with applications for percolation clusters of $ \mathbb{Z}^{d} $ and graphs which have polynomial isoperimetry and growth., Comment: 41 pages

Critical nets in $\mathbb{R}^k$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices (leaves) have a fixed position. In contrast to what happens on generic manifolds, we show that, if n is the number of 1-valent vertices, the total length of the edges not incident with a 1-valent vertex is bounded by rn (where r is the outer radius), the degree of any vertex is bounded by n and that the number of edges (and hence the number of vertices) is bounded by nl where l is related to the combinatorial diameter of the graph., Comment: 19 pages, 1 figure

The aim of the current paper is to explore the implications on the group $G$ of the non-vanishing of the cohomology in degree one of one of its representation $\pi$, given some mixing conditions on $\pi$. In one direction, harmonic cocycles are used to show that the FC-centre should be finite (for mildly mixing unitary representations). Next, for any subgroup $H

We introduce the notion of connection thickness of spheres in a Cayley graph, related to dead-ends and their retreat depth. It was well-known that connection thickness is bounded for finitely presented one-ended groups. We compute that for natural generating sets of lamplighter groups on a line or on a tree, connection thickness is linear or logarithmic respectively. We show that it depends strongly on the generating set. We give an example where the metric induced at the (finite) thickness of connection gives diameter of order $n^2$ to the sphere of radius $n$. We also discuss the rarity of dead-ends and the relationships of connection thickness with cut sets in percolation theory and with almost-convexity. Finally, we present a list of open questions about spheres in Cayley graphs., Comment: 42 pages

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in $\ell^p$ (\ie finite $p$-energy) for any $p\in [1,\infty[$ except constants (and, equivalently, that their reduced $\ell^p$ cohomology is trivial in degree one). Using similar arguments, it is also shown that many direct products of graphs (including all direct products of Cayley graphs) do not admit non-constant harmonic function with gradient in $\ell^p$. The proof relies on a theorem of Thomassen on spanning lines in squares of graphs., Comment: 6 pages

Let $\mathcal{T}_{q+1}$ be the $(q+1)$-regular tree and let $G$ be a group of automorphisms acting transitively on the vertices and on the boundary of $\mathcal{T}_{q+1}$. We give an upper bound for the growth of cocycles with values in any unitary representation of the group $G$. This bound is optimal by projecting the Haagerup cocycle onto an appropriate subspace of $\ell^{2}(E)$. We also obtain a description of functions conditionally of negative type which are unbounded.

A sign is introduced in the usual Laplacian on graphs and the corresponding analogue of the isoperimetric constant for this Laplacian is presented, i.e. a geometric quantity which enables to bound from above and below the first eigenvalue. The introduction of the sign in the Laplacian is motivated by the study of $2$-lifts of graphs and of the combinatorial Laplacian in higher degree., Comment: 10 pages

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 pages

A new proof of a Theorem of Nagnibeda and Pak on critical probabilities is given. The distinctive feature is to use a criterion of Folner which has been largely (and inexplicably) forgotten., Comment: 10 pages. The proof of Theorem 4.1 in the previous version contains a mistake; the validity of the corresponding statement is open (see question 5.5 of the current version). The current Theorem 4.1 gives Folner's original criterion, while Theorem 5.4 shows it can in a weaker guise be applied to actions

To obtain groups with bounded harmonic functions (which are not hyperbolic), one of the most frequent way is to look at some semi-direct products (\eg lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in $\ell^p$, for $p\in [1,\infty[$., Comment: 10 pages

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 figure

The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger & Gromov. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results showing its triviality are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; when $p \in ]1,\infty[$ and $G$ is Liouville (in particular, of intermediate growth). This is done by answering a question of Pansu assuming the graph satisfies an isoperimetric profile. Namely, the triviality of the reduced $\ell^p$-cohomology is equivalent to the absence of non-constant bounded (equivalently, not necessarily bounded) harmonic functions with gradient in $\ell^q$ ($q$ depends on the profile). In particular, one reduces questions of non-linear analysis ($p$-harmonic functions) to linear ones (harmonic functions with a restrictive growth condition)., Comment: 25 pages

In this paper, it is shown that the reduced $\ell^p$-cohomology is trivial for a class of finitely generated amenable groups called transport amenable. These groups are those for which there exist a sequence of measures $\xi_n$ converging to a left-invariant mean and such that the transport cost between $\xi_n$ displaced by multiplication on the right by a fixed element and $\xi_n$ is bounded in $n$. This class contains groups with controlled F{\o}lner sequence (such as polycylic groups) as well as some wreath products (such as arbitrary wreath products of finitely generated Abelian groups)., Comment: 13 pages. Added references, clarified some calculations

This article establishes more properties of the $\ell^p$ dimension introduced in a previous article. Given an amenable group $\Gamma$ acting by translation on $\ell^p(\Gamma)$, this is just a number, associated to the (usually infinite dimensional) subspaces $Y$ of $\ell^p(G)$ which are invariant under the action of $G$, satisfying dimension-like properties. As a consequence, for $p\in [1,2]$, if $Y$ is a closed non-trivial $\Gamma$-invariant subspace of $\ell^p(\Gamma;V)$ and let $Y_n$ is an increasing sequence of closed $\Gamma$-invariant subspace such that $\srl{\cup Y_n} = \ell^p(\Gamma;V)$, then there exist a $k$ such that $Y_k \cap Y \neq \{0\}$., Comment: 22 pages

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H_0,H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud-Wigner formula of quantum mechanics. Using recent results of V. Buslaev and A. Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincar\'e scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincar\'e ball., Comment: 22 pages

We consider on a symplectic manifold M with Poisson bracket {,} an Hamiltonian H with complete flow and a family Phi=(Phi_1,...,Phi_d) of observables satisfying the condition {{Phi_j,H},H}=0 for each j. Under these assumptions, we prove a new formula relating the time evolution of localisation observables defined in terms of Phi to the variation of energy along classical orbits. The correspondence between this formula and a formula established recently in the framework of quantum mechanics is put into evidence. Among other examples, our theory applies to Stark Hamiltonians, homogeneous Hamiltonians, purely kinetic Hamiltonians, the repulsive harmonic potential, the simple pendulum, central force systems, the Poincare ball model, covering manifolds, the wave equation, the nonlinear Schroedinger equation, the Korteweg-de Vries equation and quantum Hamiltonians defined via expectation values., Comment: 23 pages

The Runge approximation theorem for holomorphic maps (U -> C) is a fundamental result in complex analysis. The aim of this article is to prove such a result for (pseudo-)holomorphic maps from a compact Riemann surface to a compact (almost-)complex manifold M under certain assumptions. Though the setting is definitively that of pseudo-holomorphic maps it also covers some complex varieties., Comment: 36 pages

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 P2

Let G be a infinite countable group which acts naturally on l^p(G). We introduce a modification of mean dimension which is an obstruction for l^p(G) and l^q(G) to be H\"older conjugates., Comment: 4 pages

Gromov has shown how to construct holomorphic maps of the plane to a complex manifold with prescribed values on a lattice. In the present paper, a similar interpolation theorem for pseudo-holomorphic maps from the cylinder S to an almost-complex manifold (M,J) is proved. Properties of the space of pseudo-holomorphic maps from S to (M,J) are derived, in particular a lower bound for the size of this space (in terms of mean dimension). When M is a moduli space of curves, this gives a construction of non-compact complex surfaces. The methods involve an infinite number of surgeries. On the way, a refinement of the gluing process for two pseudo-holomorphic curves is obtained, establishing the geometric behavior of two glued pseudo-holomorphic curves., Comment: 24 pages

We say a map f:X \to Y is an \epsilon-embedding if it is continuous and the diameter of the fibres is less than \epsilon. This type of maps is used in the notion of Urysohn width (sometimes referred to as Alexandrov width), a_n(X). It is the smallest real number such that there exists an \epsilon-embedding from X to a n-dimensional polyhedron. Surprisingly few estimations of these numbers can be found, and one of the aims of this paper is to present some. Following Gromov, we take the slightly different point of view by looking at the smallest dimension n for which there exists a \epsilon-embedding to a polyhedron of dimension n. While bounds are obtained using Hadamard matrices, the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of n+1 points not contained in a hemisphere (obtained by methods very close to those of Ivanov and Pichugov). We are also able to give a complete description in dimension 3 for 1 \leq p \leq 2., Comment: 16 pages