arXiv subjects

Romain Tessera

Publications and source records attributed to Romain Tessera.

At least 19 recordsLinked to original sources

arXiv2026

Virtual splittings of right-angled Artin groups

In this article, we determine, given a finite graph $Γ$ and an integer $n \geq 1$, when a right-angled Artin group $A(Γ)$ virtually splits over an abelian subgroup of rank $n$. More precisely, we show that the following assertions are equivalent: (1) $A(Γ)$ admits $\mathbb{Z}^n$ as a codimension-one subgroup, (2) $A(Γ)$ virtually splits over $\mathbb{Z}^n$, (3) $A(Γ)$ splits over $\mathbb{Z}^n$, and (4) $Γ$ either is a complete graph with $n+1$ vertices or contains a complete subgraph of size $n$ that has a subgraph separating $Γ$.

math.GR↗

arXiv2026

Coarse separation and splittings in right-angled Artin groups

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph $Γ$, we show that $A(Γ)$ splits over an abelian subgroup if and only if it is coarsely separable by a family of subexponential growth, which amounts to saying that $Γ$ is complete or separated by a complete subgraph.

math.GR↗

arXiv2026

Coarse separation and splittings in hyperbolic groups

We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth.

math.GR↗

arXiv2026

Concentration inequalities for maximal displacement of random walks on groups of polynomial growth

We prove Gaussian concentration inequalities for maximal displacement of compactly supported random walks on a compactly generated locally compact group with polynomial growth. Concentration inequalities with different exponents hold for non-centred random walks as well, after correction by the drift. When the support of the measure generates a virtually nilpotent group, we provide an effective version of this result. These more refined estimates rely on the existence of a ``quantitative splitting'' of a virtually simply connected nilpotent group, a result which may be of independent interest. As applications, we deduce that the same concentration inequalities hold for centred random walks on the following classes of groups: amenable connected Lie groups (including non-unimodular ones), polycyclic and more generally finitely generated solvable groups with finite Prüfer rank. This shows in particular that centred random walk are diffusive on such groups. For polycyclic groups, this strengthens and completes partial results previously obtained by Russ Thompson in 2011.

math.PR↗

arXiv2025

First Passage percolation on a hyperbolic graph admits bi-infinite geodesics

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg is whether there exists a two-sided infinite geodesic in first passage percolation on Z^2, and more generally on Z^n for n>1. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite exponential moment for the random lengths, we prove that if a graph X has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on X.

math.PR↗

arXiv2025

Small doubling implies small tripling for balls of large radius

We show that if $K\ge1$ is a parameter and $S$ is a finite symmetric subset of a group containing the identity such $|S^{2n}|\le K|S^n|$ for some integer $n\ge2K^2$, then $|S^{3n}|\le\exp(\exp(O(K^2)))|S^n|$. Such a result was previously known only under the stronger assumption that $|S^{2n+1}|\le K|S^n|$. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.

math.CO↗

arXiv2025

Measure equivalence and sofic approximations

We introduce a technique for producing a measure coupling between two sofic groups from a family of maps between their sofic approximations. We exploit this to construct measure couplings between pairs of groups with prescribed integrability conditions. As an application we show that solvable Baumslag-Solitar groups, Lamplighters and the group SOL are all exponentially measure equivalent to one another: in particular they are L^p measure equivalent for all p. This is in sharp contrast with the fact that these groups are in general not quasi-isometric to one another: indeed, for instance the lamplighter with lamp group Z/3Z is not quasi-isometric the lamplighter with lamp group Z/2Z.

math.GR↗

arXiv2025

On the stability of hyperbolicity under quantitative measure equivalence

A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is $(\mathrm{L}^p,\mathrm{L}^{\infty})$. Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and $(\mathrm{L}^p,\mathrm{L}^{\infty})$, then $p$ must be less than some $p_0$ only depending on the hyperbolic group.

math.GR↗

arXiv2025

$L^p$ measure equivalence of nilpotent groups

We classify compactly generated locally compact groups of polynomial growth up to $L^p$ measure equivalence (ME) for all $p\leq 1$. To achieve this, we combine rigidity results (previously proved for discrete groups by Bowen and Austin) with new constructions of explicit orbit equivalences between simply connected nilpotent Lie groups. In particular, we prove that for every pair of simply connected nilpotent Lie groups there is an $L^p$ orbit equivalence for some $p>0$, where we can choose $p>1$ if and only if the groups have isomorphic asymptotic cones. We also prove analogous results for lattices in simply connected nilpotent Lie groups. This yields a strong converse of Austin's Theorem that two nilpotent groups which are $L^1$ ME have isomorphic Carnot graded groups. We also address the much harder problem of extending this classification to $L^p$ ME for $p>1$: we obtain the first rigidity results, providing examples of nilpotent groups with isomorphic Carnot graded groups (hence $L^1$ OE) which are not $L^p$ ME for some finite (explicit) $p$. For this we introduce a new technique, which consists of combining induction of cohomology and scaling limits via the use of a theorem of Cantrell. Finally, in the appendix, we extend theorems of Bowen, Austin and Cantrell on $L^1$ ME to locally compact groups.

math.GR↗

arXiv2024

Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs

We prove sharp bounds on the probability that the simple random walk on a vertex-transitive graph escapes the ball of radius $r$ before returning to its starting point. In particular, this shows that if the ball of radius $r$ has size slightly greater than quadratic in $r$ then this probability is bounded from below. On the other hand, we show that if the ball of radius $r$ has volume slightly less than cubic in $r$ then this probability decays logarithmically for all larger balls. These results represent a finitary refinement of Varopoulos's theorem that a random walk on a vertex-transitive graph is recurrent if and only if the graph has at most quadratic volume growth. They also imply the existence of a gap at $0$ for escape probabilities: there exists a universal constant $c>0$ such that the random walk on an arbitrary vertex-transitive graph is either recurrent or has a probability of at least $c$ of escaping to infinity. We also prove versions of these results for finite graphs, in particular confirming and strengthening a conjecture of Benjamini and Kozma from 2002. Amongst other things, we also generalise our results to give a sharp finitary version of the characterisation of $p$-parabolic vertex-transitive graphs, prove a number of sharp isoperimetric inequalities for vertex-transitive graphs, and prove a locality result for the escape probability of the random walk on a vertex-transitive graph that can be seen as an analogue of Schramm's locality conjecture for the critical percolation probability.

math.PR↗

arXiv2024

On coarse embeddings of amenable groups into hyperbolic graphs

In this note we prove that if a finitely generated amenable group admits a regular map to a direct product of a hyperbolic space and a euclidean space, then it must be virtually nilpotent. We deduce that an amenable group regularly embeds into a hyperbolic group if and only if it is virtually nilpotent, answering a question of Hume and Sisto. We describe an application to Lorentz geometry due to Charles Frances.

math.GR↗

arXiv2024

Coarse separation and large-scale geometry of wreath products

In this article, we introduce and study a natural notion of coarse separation for metric spaces, with an emphasis on coarse separation by subspaces of polynomial or subexponential growth. For instance, we show that symmetric spaces of non-compact type different from $\mathbb{H}_\mathbb{R}^2$ and thick Euclidean buildings of rank $\geq 2$ cannot be coarsely separated by subspaces of subexponential growth; and that a connected nilpotent Lie group of growth degree $D \geq 2$ cannot be coarsely separated by a subspace of polynomial degree $\leq D-2$. We apply these results to the large-scale geometry of amalgamated free products and wreath products. The latter application is based on an Embedding Theorem that generalises previous work of the last two authors, and which is of independent interest. We also discuss some further applications to the distorsion of coarse embeddings between certain metric spaces.

math.GR↗

arXiv2024

Balls in groups: volume, structure and growth

We give sharp bounds in Breuillard, Green and Tao's finitary version of Gromov's theorem on groups with polynomial growth. Precisely, we show that for every non-negative integer d there exists $c=c(d)>0$ such that if $G$ is a group with finite symmetric generating set $S$ containing the identity and $|S^n|\le cn^{d+1}|S|$ for some positive integer $n$ then there exist normal subgroups $H\leΓ\le G$ such that $H\subseteq S^n$, such that $Γ/H$ is $d$-nilpotent (i.e. has a central series of length $d$ with cyclic factors), and such that $[G:Γ]\le g(d)$, where $g(d)$ denotes the maximum order of a finite subgroup of $GL_d(\mathbb{Z})$. The bounds on both the nilpotence and index are sharp; the previous best bounds were $O(d)$ on the nilpotence, and an ineffective function of $d$ on the index. In fact, we obtain this as a small part of a much more detailed fine-scale description of the structure of $G$. These results have a wide range of applications in various aspects of the theory of vertex-transitive graphs: percolation theory, random walks, structure of finite groups, scaling limits of finite vertex-transitive graphs.... We obtain some of these applications in the present paper, and treat others in companion papers. Some are due to or joint with other authors.

math.GR↗

arXiv2024

Lamplighter-like geometry of groups

In this article, we introduce halo products as a natural generalisation of wreath products. They also encompass lampshuffler groups $\mathrm{FSym}(H) \rtimes H$ and lampcloner groups $\mathrm{FGL}(H) \rtimes H$, as well as many possible variations based for instance on braid groups, Thompson's groups, mapping class groups, automorphisms of free groups. We build a geometric framework that allows us to study the large-scale geometry of halo groups, providing refined invariants distinguishing various halo groups up to quasi-isometry.

math.GR↗

arXiv2023

Folding median graphs

Extending Stallings' foldings of trees, we show in this article that every parallel-preserving map between median graphs factors as an isometric embedding through a sequence of elementary transformations which we call foldings and swellings. This new construction proposes a unified point of view on Beeker and Lazarovich's work on folding pocsets and on Ben-Zvi, Kropholler, and Lyman's work on folding nonpositively curved cube complexes.

math.GR↗

arXiv2023

Cone-equivalent nilpotent groups with different Dehn functions

For every $k\geqslant 3$, we exhibit a simply connected $k$-nilpotent Lie group $N_k$ whose Dehn function behaves like $n^k$, while the Dehn function of its associated Carnot graded group $\mathsf{gr}(N_k)$ behaves like $n^{k+1}$. This property and its consequences allow us to reveal three new phenomena. First, since those groups have uniform lattices, this provides the first examples of pairs of finitely presented groups with bilipschitz asymptotic cones but with different Dehn functions. The second surprising feature of these groups is that for every even integer $k \geqslant 4$ the centralized Dehn function of $N_k$ behaves like $n^{k-1}$ and so has a different exponent than the Dehn function. This answers a question of Young. Finally, we turn our attention to sublinear bilipschitz equivalences (SBE). Introduced by Cornulier, these are maps between metric spaces inducing bi-Lipschitz homeomorphisms between their asymptotic cones. These can be seen as weakenings of quasiisometries where the additive error is replaced by a sublinearly growing function $v$. We show that a $v$-SBE between $N_k$ and $\mathsf{gr}(N_k)$ must satisfy $v(n)\succcurlyeq n^{1/(2k + 2)}$, strengthening the fact that those two groups are not quasiisometric. This is the first instance where an explicit lower bound is provided for a pair of SBE groups.

math.GR↗

arXiv2023

Belinskaya's theorem is optimal

Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an $\mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all $p<1$ the integrability condition on the cocycle cannot be relaxed to being in $\mathrm{L}^p$. This also allows us to answer a question of Kerr and Li: for ergodic measure-preserving transformations, Shannon orbit equivalence doesn't boil down to flip-conjugacy.

math.DS↗

arXiv2023

Geometric amenability in totally disconnected locally compact groups

We give a short geometric proof of a result of Soardi & Woess and Salvatori that a quasitransitive graph is amenable if and only if its automorphism group is amenable and unimodular. We also strengthen one direction of that result by showing that if a compactly generated totally disconnected locally compact group admits a proper Lipschitz action on a bounded-degree amenable graph then that group is amenable and unimodular. We pass via the notion of geometric amenability of a locally compact group, which has previously been studied by the second author and is defined by analogy with amenability, only using right Folner sets instead of left Folner sets. We also introduce a notion of uniform geometric non-amenability of a locally compact group, and relate this notion in various ways to actions of that group on graphs and to its modular homomorphism.

math.GR↗