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Amaury Freslon

Publications and source records attributed to Amaury Freslon.

At least 19 recordsLinked to original sources
arXiv2026

Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras

We compute the quantum isometry groups of Cuntz-Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups. Contrary to the classical isometry groups, whose actions on the Cuntz-Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra. This also leads to the construction of a (genuinely quantum) ergodic action of a compact matrix quantum group on the Cantor space.

math.OA↗
arXiv2025

Block structures of graphs and quantum isomorphism

We prove that for every pair of quantum isomorphic graphs, their block trees and their block graphs are isomorphic, and that such an isomorphism can be chosen so that the corresponding blocks are quantum isomorphic -- in particular, 2-connectedness is preserved under quantum isomorphism. We conclude with some corollaries, including obtaining some necessary conditions on a pair of quantum isomorphic, not isomorphic graphs with a minimal number of vertices.

math.CO↗
arXiv2025

Free Inhomogeneous Wreath Product of Quantum Groups

We introduce the free inhomogeneous wreath product of compact matrix quantum groups, which generalizes the free wreath product (Bichon 2004). We use this to present a general technique to determine quantum automorphism groups of connected graphs in terms of their maximal biconnected subgraphs, provided that we have sufficient information about their quantum automorphism groups. We show that this requirement is met for forests, outerplanar graphs, and block graphs leading to algorithms to compute the quantum automorphism groups of these graphs.

math.QA↗
arXiv2025

Gaussian Generating functionals on easy quantum groups

We describe all Gaussian generating functionals on several easy quantum groups given by non-crossing partitions. This includes in particular the free unitary, orthogonal and symplectic quantum groups. We further characterize central Gaussian generating functionals and describe a centralization procedure yielding interesting (non-Gaussian) generating functionals.

math.QA↗
arXiv2025

Noncommutative properties of 0-hyperbolic graphs

We study several noncommutative properties of 0-hyperbolic graphs. In particular, we prove that 0-hyperbolicity is preserved under quantum isomorphism. We also compute the quantum automorphism groups of 0-hyperbolic graphs and characterise the ones with quantum symmetry.

math.CO↗
arXiv2025

Tracial central states on compact quantum groups

Motivated by classical investigation of conjugation invariant positive-definite functions on discrete groups, we study tracial central states on universal C*-algebras associated with compact quantum groups, where centrality is understood in the sense of invariance under the adjoint action. We fully classify such states on q-deformations of compact Lie groups, on free orthogonal quantum groups, quantum permutation groups and on quantum hyperoctahedral groups.

math.OA↗
arXiv2024

Discrete quantum subgroups of free unitary quantum groups

We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups $U_{N}^{+}$. In other words, we classify all discrete quantum subgroups of $\widehat{U}_{N}^{+}$, thereby proving a quantum variant of Kurosh's theorem to some extent. This yields interesting families which can be described using free wreath products and free complexifications. They can also be seen as quantum automorphism groups of specific quantum graphs which generalize finite rooted regular trees, providing explicit examples of quantum trees.

math.OA↗
arXiv2024

Classical actions of quantum permutation groups

We describe explicitly all actions of the quantum permutation groups on classical compact spaces. In particular, we show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated to non-crossing partitions.

math.OA↗
arXiv2023

Connectedness and Gaussian Parts for Compact Quantum Groups

We introduce the Gaussian part of a compact quantum group $\mathbb{G}$, namely the largest quantum subgroup of $\mathbb{G}$ supporting all the Gaussian functionals of $\mathbb{G}$. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and $q$-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.

math.QA↗
arXiv2023

Tannaka-Krein reconstruction and ergodic actions of easy quantum groups

We give a new alternative version of the reconstruction procedure for ergodic actions of compact quantum groups and we refine it to include characterizations of (braided commutative) Yetter-Drinfeld C*-algebras. We then use this to construct families of ergodic actions of easy quantum groups out of combinatorial data involving partitions and study them. Eventually, we use this categorical point of view to show that the quantum permutation group cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

math.OA↗
arXiv2021

Free wreath products with amalgamation

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name "free wreath products of pairs" in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.

math.QA↗
arXiv2021

Cutoff profiles for quantum Lévy processes and quantum random transpositions

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.

math.PR↗
arXiv2020

Torsion and K-theory for some free wreath products

We classify torsion actions of free wreath products of arbitrary compact quantum groups and use this to prove that if $\mathbb{G}$ is a torsion-free compact quantum group satisfying the strong Baum-Connes property, then $\mathbb{G}\wr_{\ast}S_{N}^{+}$ also satisfies the strong Baum-Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by $SO_{q}(3)$.

math.OA↗
arXiv2019

Positive definite functions and cut-off for discrete groups

We consider the sequence of powers of a positive definite function on a discrete group. Taking inspiration from random walks on compact quantum groups, we give several examples of situations where a cut-off phenomenon occurs for this sequence with respect to the operator norm of the regular representation, including free groups and infinite Coxeter groups. We also give examples of absence of cut-off using free groups again.

math.GR↗
arXiv2018

Topological generation and matrix models for quantum reflection groups

We establish several new topological generation results for the quantum permutation groups $S^+_N$ and the quantum reflection groups $H^{s+}_N$. We use these results to show that these quantum groups admit sufficiently many "matrix models". In particular, all of these quantum groups have residually finite discrete duals (and are, in particular, hyperlinear), and certain "flat" matrix models for $S_N^+$ are inner faithful.

math.OA↗
arXiv2018

Quantum reflections, random walks and cut-off

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of such reflections. We also study random walks on quantum reflection groups and more generally free wreath products of finite group by quantum permutation groups.

math.PR↗
arXiv2018

On two-coloured noncrossing quantum groups

We classify compact quantum groups associated to noncrossing partitions coloured with two elements $x$ and $y$ which are their own inverses. Together with the work of P. Tarrago and M. Weber, this completes the classification of all noncrossing quantum groups on two colours. We also give some general structure results concerning noncrossing quantum groups and suggest some more general classification statements.

math.QA↗