arXiv subjects
Publications and source records attributed to Antoine Gournay.
The Liouville property is a strong form of amenability, but contrary to amenability, it is not well-behaved under extensions. In this paper it is shown that, for some measures, the Liouville property is preserved by [FC-]hypercentral extensions. To this end a projection from $\ell^\infty$ onto the space of harmonic functions is introduced.
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 $\ell^p$.
In this paper we show that groups for which the probability of return of a random walk is bounded below by $K_1 exp(-K_2n^c)$ have no non-constant harmonic functions with gradient in $\ell^p$. The proof relies on results from $\ell^p$-cohomology, a form of radial isoperimetry, transport patterns and revisiting some results of Følner.
We prove that the lamplighter group admits an injective Lipschitz map to any finitely generated metabelian group which is not virtually nilpotent. This implies that finitely generated metabelian groups satisfy the ``analytically thin/analytically thick'' dichotomy recently introduced by Hume, Mackay and Tessera.
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).
We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) 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.
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.
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.
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 $π$, given some mixing conditions on $π$. 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 $H<G$, $H$ will either be "small", almost-malnormal or $π_{|H}$ also has non-trivial cohomology in degree one (in this statement, "small", reduced vs unreduced cohomology and unitary vs generic depend on the mixing condition). The notion of q-normal subgroups is an important ingredient of the proof and results on the vanishing of the reduced $\ell^p$-cohomology in degree one are obtained as an intermediate step.
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.
Lower bound on the equivariant Hilbertian compression exponent $α$ are obtained using random walks. More precisely, if the probability of return of the simple random walk is $\succeq \textrm{exp}(-n^γ)$ in a Cayley graph then $α\geq (1-γ)/(1+γ)$. This motivates the study of further relations between return probability, speed, entropy and volume growth. For example, if $|B_n| \preceq e^{n^ν}$ then the speed is $\preceq n^{1/(2-ν)}$. Under a strong assumption on the off-diagonal decay of the heat kernel, the lower bound on compression improves to $α\geq 1-γ$. 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 $γ<1/2$.
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 $p\in [1,\infty[$.
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.
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).
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.
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.
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 $ξ_n$ converging to a left-invariant mean and such that the transport cost between $ξ_n$ displaced by multiplication on the right by a fixed element and $ξ_n$ is bounded in $n$. This class contains groups with controlled Følner sequence (such as polycylic groups) as well as some wreath products (such as arbitrary wreath products of finitely generated Abelian groups).
This article establishes more properties of the $\ell^p$ dimension introduced in a previous article. Given an amenable group $Γ$ acting by translation on $\ell^p(Γ)$, 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 $Γ$-invariant subspace of $\ell^p(Γ;V)$ and let $Y_n$ is an increasing sequence of closed $Γ$-invariant subspace such that $\srl{\cup Y_n} = \ell^p(Γ;V)$, then there exist a $k$ such that $Y_k \cap Y \neq \{0\}$.