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Pierre-Emmanuel Caprace

Publications and source records attributed to Pierre-Emmanuel Caprace.

At least 19 recordsLinked to original sources
arXiv2026

Ascending chains of irreducible lattices, bi-reversible automata and affine arithmetic groups

For each $n \geq 2$, we construct an ascending chain of irreducible lattices in the product of $n$ homogeneous trees. Moreover, for each pair of integers $m_1, m_2 \geq 1$, we define explicitly a bi-reversible automaton $\mathcal B$ such that the group $G_{\mathcal B}$ defined by the automaton $\mathcal B$ has finiteness length $m_1$ (i.e. it is of type $\mathrm{F}_{m_1}$ but not of type $\mathrm{FP}_{m_1+1}$), and the group $G_{\mathcal B^*}$ defined by the dual automaton has finiteness length $m_2$. Both constructions rely on the consideration of $S$-arithmetic groups in the affine group of a global function field.

math.GR↗
arXiv2026

On the Howe--Moore property for automorphism groups of buildings

Let $G$ be a closed type-preserving subgroup of the automorphism group of a thick locally finite building $X$ of finite rank, and assume that $G$ acts Weyl-transitively. We prove that every unitary representation of $G$ is mixing, unless its restriction to a parabolic subgroup of minimal non-spherical type is amenable in the sense of Bekka. It follows that every unitary representation of $G$ that is weakly contained in the regular representation, is mixing. In case $X$ is of minimal non-spherical type and its thickness satisfies some modest lower bound, we deduce that $G$ has the Howe--Moore property provided its only compact quotient is trivial. We also obtain results on rigidity of invariant random subgroups for Kac--Moody lattices of compact hyperbolic type, yielding examples of infinite finitely presented Kazhdan groups with exactly two ergodic invariant random subgroups.

math.GR↗
arXiv2026

Fractal anti-tori

Let $\Gamma$ be a group acting properly and cocompactly on the product of two trees $T_1$ and $T_2$. An anti-torus is a non-periodic flat plane in $T_1 \times T_2$ that is the convex hull of two secant periodic lines. That notion was introduced by Dani Wise as a tool to show that $\Gamma$ is irreducible. We establish a new criterion ensuring the existence of anti-tori, and use it to prove that if $\Gamma$ is an $S$-arithmetic lattice in a product of simple algebraic groups of rank one, then $T_1\times T_2$ contains anti-tori. As a byproduct, we obtain a sufficient condition ensuring that a group defined by a bi-reversible automaton contains non-abelian free sub-semigroups. We also introduce a new class of irreducible lattices acting regularly on the vertex set of a product of two trees, and containing anti-tori that are fractal aperiodic tilings of the plane. This establishes a connection between lattices in products of trees and substitution tilings.

math.GR↗
arXiv2026

On the self-similarity of rational power series with matrix coefficients

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that $Q$ is invertible in $ A[\![x_1, \dots, x_n]\!]$. Let also $\mathcal M \colon \mathbf Z^n \to A$ be the map associating to the $n$-tuple of integers $(\alpha_1, \dots, \alpha_n)$ the coefficient of the monomial $x_1^{\alpha_1} \dots x_n^{\alpha_n}$ in the development of the rational fraction $PQ^{-1}$ as a power series (the support of $\mathcal M$ is contained in $\mathbf N^n$). Our main result ensures that the map $\mathcal M$, viewed as a tiling of $\mathbf R^n$ by unit cubes with color set $A$, is self-similar. The self-similarity is expressed in terms of invariance under substitutions. By specializing to $d=1$, $n=2$, $P=1$ and $Q =1-x_1-x_2$, we recover the well-known self-similarity feature of the binomial coefficients modulo $p$.

math.CO↗
arXiv2026

Center-preserving irreducible representations of finite groups

Given finite groups $H \leq G$, a representation $\sigma$ of $G$ is called center-preserving on $H$ if the only elements of $H$ that become central under $\sigma$ are those that were already central in $G$. We prove that if $H$ has a faithful irreducible representation $\rho$, then at least one of the irreducible components of the induction $\operatorname{Ind}_H^G(\rho)$ is center-preserving on $H$. In consequence, $H$ has a faithful irreducible representation if and only if every finite group $G$ containing $H$ as a subgroup has an irreducible representation whose restriction to $H$ is faithful, and which is center-preserving on $H$. In addition, we give examples illustrating the sharpness of the statement, and discuss the connection with projective representations.

math.GR↗
arXiv2025

The type I dichotomy for two-step nilpotent locally compact groups

We address the type I dichotomy for two-step nilpotent locally compact groups. Invoking work of Baggett-Kleppner, we characterize the closed points of the unitary dual of such a group $G$ purely in terms of the group structure. An algebraic criterion characterizing when $G$ is a type I group is derived. We show that this criterion automatically holds if $G$ is a central extension of vector groups over a non-discrete locally compact field $k$ such that the commutator map is $k$-bilinear. As an application, we show that the unipotent radicals of minimal parabolics in simple algebraic groups of $k$-rank one are type I groups. We also discuss the type I dichotomy for $p$-torsion contraction groups, and exhibit, for each prime $p$, uncountably many pairwise non-isomorphic such groups that are not type I. This answers a recently posed question by the second author. Finally, we adapt a recent construction of Chirvasitu to obtain numerous examples of two-step nilpotent torsion locally compact groups that are not type I, but that embed as closed cocompact normal subgroups in two-step nilpotent groups that are type I.

math.RT↗
arXiv2024

On compact uniformly recurrent subgroups

Let a group $Γ$ act on a paracompact, locally compact, Hausdorff space $M$ by homeomorphisms and let $2^M$ denote the set of closed subsets of $M$. We endow $2^M$ with the Chabauty topology, which is compact and admits a natural $Γ$-action by homeomorphisms. We show that for every minimal $Γ$-invariant closed subset $\mathcal Y$ of $2^M$ consisting of compact sets, the union $\bigcup \mathcal{Y}\subset M$ has compact closure. As an application, we deduce that every compact uniformly recurrent subgroup of a locally compact group is contained in a compact normal subgroup. This generalizes a result of Ušakov on compact subgroups whose normalizer is compact.

math.GR↗
arXiv2024

Amalgams of rational unipotent groups and residual nilpotence

We provide sufficient conditions for a free amalgamated product of torsionfree nilpotent groups to be residually nilpotent. We also characterise the residual nilpotence of certain higher-dimensional amalgams of unipotent groups over the rationals (known as KMS groups) in terms of their defining Cartan matrix. As an application, we give a normal form for the elements of a minimal Kac-Moody group over the rationals.

math.GR↗
arXiv2024

Lattice envelopes of right-angled Artin groups

Let $\Gamma$ be a finite simplicial graph with at least two vertices, and let $G(\Gamma)$ be the associated right-angled Artin group. We describe a locally compact group $\mathcal U$ containing $G(\Gamma)$ as a cocompact lattice. If $\Gamma$ is not a join (i.e. the complement graph is connected), the group $\mathcal U$ is non-discrete, almost simple, but not virtually simple: it has a smallest normal subgroup $\mathcal U^+$ which is an open simple subgroup, and the quotient $\mathcal U/\mathcal U^+$ is isomorphic to the right-angled Coxeter group $W(\Gamma)$. Under suitable assumptions on $\Gamma$, we rely on work by Bader-Furman-Sauer and Huang-Kleiner to show that $\mathcal U \rtimes \mathrm{Aut}(\Gamma)$ is the universal lattice envelope of $G(\Gamma)$: for every lattice envelope $H$ of $G(\Gamma)$, there is a continuous proper homomorphism $H \to \mathcal U \rtimes \mathrm{Aut}(\Gamma)$. In particular, no lattice envelope of $G(\Gamma)$ is virtually simple. We also show that no locally compact group quasi-isometric to $G(\Gamma)$ is virtually simple. This contrasts with the case of free groups. The group $\mathcal U$ is a universal automorphism group of the Davis building of $G(\Gamma)$, with prescribed local actions. As an application, we describe the algebraic structure of the full automorphism group of the Cayley graph of $G(\Gamma)$ with respect to its standard generating set.

math.GR↗
arXiv2023

Growing trees from compact subgroups

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups $G$, finding a sufficient condition for $G$ to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group $G/N$ on a tree with faithful micro-supported action on the boundary, where $N$ is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of $G/N$ with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group $G$, for all direct factors of open subgroups of $G$ to be trivial or open.

math.GR↗
arXiv2023

Tame automorphism groups of polynomial rings with property (T) and infinitely many alternating group quotients

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of $\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of $\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ on the $n$-dimensional affine spaces over finite extensions of $\mathbf F_p$. As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree $4$ for alternating groups of degree $p^7-1$ for any odd prime $p$.

math.GR↗
arXiv2022

Piecewise strongly proximal actions, free boundaries and the Neretin groups

A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary representations that are weakly equivalent. This implies that the Neretin groups are not of type I, thereby answering a question of Y.~Neretin.

math.GR↗
arXiv2022

Locally normal subgroups and ends of locally compact Kac-Moody groups

A locally normal subgroup in a topological group is a subgroup whose normaliser is open. In this paper, we provide a detailed description of the large-scale structure of closed locally normal subgroups of complete Kac-Moody groups over finite fields. Combining that description with the main result from arXiv:2111.07066, we show that under mild assumptions, if the Kac-Moody group is one-ended (a property that is easily determined from the generalised Cartan matrix), then it is locally indecomposable, which means that no open subgroup decomposes as a nontrivial direct product.

math.GR↗
arXiv2022

A type I conjecture and boundary representations of hyperbolic groups

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Ł. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

math.GR↗
arXiv2021

A radius 1 irreducibility criterion for lattices in products of trees

Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 \geq 3$. Let also $Γ\leq \mathrm{Aut}(T_1) \times \mathrm{Aut}(T_2)$ be a group acting freely and transitively on $VT_1 \times VT_2$. For $i=1$ and $2$, assume that the local action of $Γ$ on $T_i$ is $2$-transitive; if moreover $d_i \geq 7$, assume that the local action contains $\mathrm{Alt}(d_i)$. We show that $Γ$ is irreducible, unless $(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $Γ$ that can be checked purely in terms of its local action on a ball of radius~$1$ in $T_1$ and $T_2$. Under the same hypotheses, we show moreover that if $Γ$ is irreducible, then it is hereditarily just-infinite, provided the local action on $T_i$ is not the affine group $\mathbf F_5 \rtimes \mathbf F_5^*$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.

math.GR↗
arXiv2021

Commensurated subgroups and micro-supported actions

Let $Γ$ be a finitely generated group and $X$ be a minimal compact $Γ$-space. We assume that the $Γ$-action is micro-supported, i.e. for every non-empty open subset $U \subseteq X$, there is an element of $Γ$ acting non-trivially on $U$ and trivially on the complement $X \setminus U$. We show that, under suitable assumptions, the existence of certain commensurated subgroups in $Γ$ yields strong restrictions on the dynamics of the $Γ$-action: the space $X$ has compressible open subsets, and it is an almost $Γ$-boundary. Those properties yield in turn restrictions on the structure of $Γ$: $Γ$ is neither amenable nor residually finite. Among the applications, we show that the (alternating subgroup of the) topological full group associated to a minimal and expansive Cantor action of a finitely generated amenable group has no commensurated subgroups other than the trivial ones. Similarly, every commensurated subgroup of a finitely generated branch group is commensurate to a normal subgroup; the latter assertion relies on an appendix by Dominik Francoeur, and generalizes a result of Phillip Wesolek on finitely generated just-infinite branch groups. Other applications concern discrete groups acting on the circle, and the centralizer lattice of non-discrete totally disconnected locally compact (tdlc) groups. Our results rely, in an essential way, on recent results on the structure of tdlc groups, on the dynamics of their micro-supported actions, and on the notion of uniformly recurrent subgroups.

math.GR↗