arXiv subjects

Corentin Le Bars

Publications and source records attributed to Corentin Le Bars.

10 recordsLinked to original sources

arXiv2026

Selflessness, MIF and opposition in groups acting on exotic buildings

We prove that groups acting freely and cocompactly on (possibly exotic) affine buildings of type $\tilde{A}_2$ and $\tilde C_2$ are mixed-identity free and have selfless reduced $C^*$-algebras. These results follow from a strong form of ping-pong dynamics that we call 'transversal contractivity'. Our main geometric result regards domesticity properties of elements in the associated polygons at infinity: we prove that, in our context, the opposite geometry of a hyperbolic element is topologically large.

math.GR↗

arXiv2026

The noncommutative topological factor theorem for rank-one product lattices

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that the corresponding classification for the full flag action of $\operatorname{SL}_3(\mathbb Z)$ would imply ordinary ITAP.

math.OA↗

arXiv2026

The geometry of triples of antipodal ideal chambers of affine buildings

In this article, we investigate two notions of genericity for triples of antipodal ideal chambers in a locally finite affine building $X$: one defined at the ideal boundary $X^{\infty}$, which we call ideal-genericity, and the other defined from within the affine building $X$, which we call affine-genericity. While ideal-genericity implies affine-genericity, the latter is the more suitable notion for constructing a barycenter map associated with affine-generic triples of antipodal ideal chambers. This perspective also allows us to establish that this barycenter map is locally constant, and hence continuous. Finally, we provide sufficient geometric conditions on the affine Weyl group associated with $X$ that guarantee both ideal- and affine-genericity for all triples of antipodal ideal chambers of $X^\infty$. These conditions yield an algorithmic method for constructing geometric configurations in $X \cup X^{\infty}$ (when they exist) that correspond to non-generic triples of antipodal ideal chambers of $X^{\infty}$. Furthermore, our computations for the irreducible finite Weyl groups of rank at least two show that automatic ideal-genericity holds for all triples of antipodal ideal chambers only in types $B_2 = C_2$, $G_2$ and $B_3$.

math.GR↗

arXiv2026

Dynamical boundaries of affine buildings: C*-simplicity and Poisson boundaries

We investigate a class of groups acting on possibly exotic affine buildings $X$ and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of chambers at infinity of $X$. For a group $G$ of general type, we prove C*-simplicity by showing that its flag limit set $Λ_{\mathcal F}(G)$ is topologically free, minimal, and strongly proximal. When $Λ_{\mathcal F}(G)$ intersects all Schubert cells relative to a limit chamber, then it is a mean proximal space, in the sense that it carries a unique proximal stationary measure for any admissible probability measure on the acting group. Lattices are established as examples of groups of general type, and their Poisson boundaries are identified. The arguments rely on constructing an equivariant barycenter map from triples of chambers in generic position to the affine building.

math.GR↗

arXiv2025

Stationary measures and random walks on $\tilde{A}_2$-buildings

We consider a non-elementary group action $G \curvearrowright X$ of a locally compact second countable group $G$ on a possibly exotic non-discrete affine building $X$ of type $\tilde{A}_2$. We prove that if $μ$ is an admissible symmetric probability measure on $G$, there is a unique $μ$-stationary measure supported on the chambers of the spherical building at infinity. We use this result to study random walks induced by the $G$-action, and we prove that if $μ$ has finite second moment, $(Z_n o)$ converges almost surely to a regular point of the boundary and the Lyapunov spectrum of the random walk is simple. Applied to Bruhat-Tits buildings, these results extend some classical theorems due to H.~Furstenberg.

math.GR↗

arXiv2025

Ergodic cocycles in hyperbolic and Hadamard spaces

We consider discrete random dynamical systems induced by a non-elementary group action on a non-proper hyperbolic space. We prove that if the system is ergodic and satisfies the ``asymptotic past and future independence condition'' as defined by Bader and Furman, the associated ergodic cocycle converges to the Gromov boundary almost surely. If the cocycle has finite first moment, we show that its drift is positive. Using hyperbolic models introduced by Petyt-Spriano-Zalloum, we prove analogous statements for groups acting on Hadamard spaces with a pair of contracting elements.

math.DS↗

arXiv2024

A Tits alternative for $\mathbb{R}$-buildings of type $\tilde{A}_2$

Let $G$ be a group with a non-elementary action on a (not necessarily discrete) $\tilde{A}_2$-buildings. We prove that, given a random walk on $G$, isometries in $G$ are strongly regular hyperbolic with high probability. As a consequence, we prove a Tits alternative for $G$, as well as a local-to-global fixed point result. We also prove that isometries of (not necessarily complete) $\mathbb{R}$-buildings are semi-simple.

math.GR↗

arXiv2024

Central limit theorem on CAT(0) spaces with contracting isometries

Let $G$ be a group with a non-elementary action on a proper CAT(0) space $X$, and let $μ$ be a measure on $G$ such that the random walk $(Z_n)_n$ generated by $μ$ has finite second moment on $X$. Let $o$ be a basepoint in $X$, and assume that there exists a rank one isometry in $G$. We prove that in this context, $(Z_n o )_n$ satisfies a Central Limit Theorem, namely that the random variables $\frac{1}{\sqrt{n}}(d(Z_n o, o) - n λ) $ converge in law to a non-degenerate Gaussian distribution $N_μ$, for $λ$ the (positive) drift of the random walk. The strategy relies on the use of hyperbolic models introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of curve graphs and cubical hyperplanes for the class of CAT(0) spaces. As a side result, we prove that the probability that the nth-step $Z_n$ acts on $X$ as a contracting isometry goes to 1 as $n$ goes to infinity.

math.GR↗

arXiv2023

Marches aléatoires et éléments contractants sur des espaces CAT(0)

This thesis is dedicated to random walks on spaces with non-positive curvature. In particular, we study the case of group actions on CAT(0) spaces that admit contracting elements, that is, whose properties mimic those of loxodromic isometries in Gromov-hyperbolic spaces. In this context, we prove several limit laws, among which the almost sure convergence to the boundary without moment assumption, positivity of the drift and a central limit theorem. In a second part, we study boundary maps and stationary measures on affine buildings of type $\tilde{A}_2$, and we show that there always exists a hyperbolic isometry for a non-elementary action by isometries on such a space. Our approach involves the use of hyperbolic models for CAT(0) spaces, which were constructed by H.~Petyt, D.~Spriano and A.~Zalloum, and measured boundary theory, whose principles come from H.~Furstenberg.

math.GR↗

arXiv2022

Random walks and rank one isometries on CAT(0) spaces

Let $G$ be a discrete group, $μ$ a measure on $G$ and $X$ a proper CAT(0) space. We show that if $G$ acts non-elementarily with a rank one element on $X$, then the pushforward $\{Z_n o \}_n$ to $X$ of the random walk generated by $μ$ converges almost surely to a rank one point of the boundary. We also show that in this context, there is a unique stationary measure on the visual boundary $\partial_\infty X$ of $X$, and that the drift of the random walk is almost surely positive.

math.GR↗