Articles

22. Chabauty limits of groups of involutions in SL(2,F)for local fields, https://arxiv.org/abs/2208.12247

with Arielle Leitner

We classify Chabauty limits of groups fixed by various (abstract) involutions over SL(2,F), where F is a finite field-extension of Qp, with p≠2. To do so, we first classify abstract involutions over SL(2,F) with F a quadratic extension of Qp, and prove p-adic polar decompositions with respect to various subgroups of p-adic SL2. Then we classify Chabauty limits of: SL(2,F) inside SL(2,E), where E is a quadratic extension of F, of SL(2,R) inside SL(2,C), and of H_θ⊂SL(2,F), where H_θ is the fixed point group of an F-involution θ over SL(2,F).

21. Polyhedral Compactifications II

with Linus Kramer and Petra Schwer, in preparation.

20. Strong transitivity, the Moufang condition and the Howe--Moore property, https://arxiv.org/abs/2011.12921, to appear in Transformation Groups

Firstly, we prove that every closed subgroup H of type-preserving automorphisms of a locally finite thick affine building \Delta of dimension >=2 that acts strongly transitive on \Delta is Moufang. If moreover \Delta is irreducible and H is topologically simple, then we show H is the isotropic simple algebraic group over a non-Archimedean local field associated with \Delta. Secondly, we generalizes the proof given in Burger--Mozes 2000 for the case of bi-regular trees to any locally finite thick affine building \Delta, and proves that any topologically simple, closed, strongly transitive and type-preserving subgroup of Aut(\Delta) has the Howe--Moore property.

19. The universal group of Burger--Mozes and the Howe--Moore property, https://arxiv.org/abs/1612.09427

By constructing a new unitary representation we prove the universal group U(F)^+ of Burger–Mozes does not have the Howe–Moore property when F is primitive but not 2-transitive. It is well known U(F)^+ does have this property when F is 2-transitive. Along the way, we give a characterization of the universal group, when F is primitive, to have the Howe–Moore property, and also prove U(F)^+ has the relative Howe–Moore property. These two results are a consequence of a strengthening of Mautner’s phenomenon for locally compact groups acting on d-regular trees and having Tits’ independence property.

18. Polyhedral Compactifications I

with Linus Kramer and Petra Schwer https://arxiv.org/abs/2002.12422.

In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

17. (Non)-escape of mass and equidistribution for horospherical actions on trees, with Vladimir Finkelshtein and Cagri Sert, Mathematische Zeitschrift, 2021, https://doi.org/10.1007/s00209-021-02852-1

Let G be a large group acting on a biregular tree T and Γ ≤ G a geometrically finite lattice. In an earlier work, the authors classified orbit closures of the action of the horospherical subgroups on G/Γ. In this article we show that, in fact, the dense orbits equidistribute to the Haar measure on G/Γ. In particular, there is no escape of mass to infinity. On the other hand, we show that new dynamical phenomena for horospherical actions appear on quotients by non-geometrically finite lattices: we give examples of non-geometrically finite lattices where an escape of mass phenomenon occurs and where the orbital averages along a Folner sequence do not converge. In the last part, as a by-product of our methods, we show that projections to Γ/T of the uniform distributions on large spheres in the tree T converge to a natural probability measure on Γ/T. Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.

16. Consistency and asymptotic normality of M-estimates of scatter on Grassmann manifolds, with Christian Mazza, https://doi.org/10.1016/j.jmva.2022.104998, Journal of Multivariate Analysis, Volume 190, July 2022

We consider data from the Grassmann manifold G(m, r) of all vector subspaces of dimension r of R^m . Building on the Grassmannian geometrical model and results from our first paper in this series, we prove here important statistical properties of M-estimates of scatter on G(m, r). Recall, canonical Grassmannian distributions G_Σ on G(m, r) are indexed by parameters Σ from the manifold M = Pos _sym ^1 (m) of positive definite symmetric matrices of determinant 1. M-estimates of scatter (GE) for general probability measures P on G(m, r) are maximizers of the Grassmannian log-likelihood −l_P (Σ) as function of Σ. Firstly, we give an explicite proof for a necessary and sufficient condition that implies existence and unicity of those GEs. Secondly, we prove a LLN like results. And thirdly, we provide a central limit theorem for a corrsponding rescaled process.

15. Geometrical properties of M-estimates of scatter on Grassmann manifolds, with Christian Mazza https://arxiv.org/abs/1812.11605, submitted.

We consider data from the Grassmann manifold G(m, r) of all vector subspaces of dimension r of R^m , and focus on the Grassmannian geometrical model targeting statistical properties studied in our second paper. Canonical Grassmannian distributions G_Σ on G(m, r) are indexed by parameters Σ from the manifold M = Pos_sym^1 (m) of positive definite symmetric matrices of determinant 1. Robust M-estimates of scatter (GE) for general probability measures P on G(m, r) are studied. Such estimators are defined to be the maximizers of the known Grassmannian log-likelihood as function of Σ. One of the novel features of this work is a strong use of the fact that M is a CAT(0) space with known visual boundary at infinity ∂M, allowing us to study existence and unicity of GEs. Along the way it is proved that the log-likelihood is convex, and under further conditions even strictly convex, when evaluated on geodesics γ (t) of M. We also recall that the sample space G(m, r) is a part of ∂M, show the distributions G Σ are SL(m, R)–quasi-invariant, and that the log-likelihood is a weighted Busemann function. Those results are heavily used in our second paper where important statistical properties (e.g., LLN and CLT) are proved.

14. Chabauty Limits of Cartan Subgroups of SL(n, Q p), with Arielle Leitner and Alain Valette, Journal of Algebra, April 2022, https://doi.org/10.1016/j.jalgebra.2021.11.032

Let C be the subgroup of all diagonal matrices in SL(n,Q_p ). In the first part of this paper we study and give a classification of the Chabauty limits of SL(n,Q_p)-conjugates of C using the action of SL(n,Q_p ) on its associated Bruhat–Tits building. Along the way we construct an explicit homeomorphism between the Chabauty compactification in sl(n,Q_p ) of SL(n,Q_p )-conjugates of the p-adic Lie algebra of C and the Chabauty compactification of SL(n,Q_p )-conjugates of C. In the second part of the paper we compute all of the Chabauty limits for n ≤ 4 (up to conjugacy). In contrast, for n ≥ 7 we prove there are infinitely many SL(n,Q_p)-nonconjugate Chabauty limits.

13. Measure rigidity for horospherical subgroups of groups acting on trees,

with Vladimir Finkelshtein and Cagri Sert, International Mathematics Research Notices, Volume 2021, Issue 21, November 2021, 16227–16270, https://doi.org/10.1093/imrn/rnz275

We prove analogues of some of the classical results in homogeneous dynamics in nonlinear setting. Let G be a

closed subgroup of the group of automorphisms of a biregular tree and \Gamma < G a discrete subgroup. For a large class of groups G, we give a classi.cation of the probability measures on G/ \Gamma invariant under horospherical subgroups. When \Gamma is a cocompact lattice, we show the unique ergodicity of the horospherical action. We prove Hedlund's theorem for geometrically .nite quotients. Finally, we show equidistribution of large compact orbits.

12. Mean field repulsive Kuramoto models: Phase locking and spatial signs, with Linard Hoessly, Christian Mazza, Xavier Richard, https://arxiv.org/abs/1803.02647, submitted.

The phenomenon of self-synchronization in populations of oscillatory units appears naturally in neurosciences. However, in some situations, the formation of a coherent state is damaging. In this article we study a repulsive mean-field Kuramoto model that describes the time evolution of n points on the unit circle, which are transformed into incoherent phase-locked states. It has been recently shown that such systems can be reduced to a three-dimensional system of ordinary differential equations, whose mathematical structure is strongly related to hyperbolic geometry. The orbits of the Kuramoto dynamical system are then described by a flow of Möbius transformations. We show this underlying dynamic performs statistical inference by computing dynamically M-estimates of scatter matrices. We also describe the limiting phase-locked states for random initial conditions using Tyler's transformation matrix. Moreover, we show the repulsive Kuramoto model performs dynamically not only robust covariance matrix estimation, but also data processing: the initial configuration of the n points is transformed by the dynamic into a limiting phase-locked state that surprisingly equals the spatial signs from nonparametric statistics. That makes the sign empirical covariance matrix to equal 1/2 Id_2, the variance-covariance matrix of a random vector that is uniformly distributed on the unit circle.

11. Chabauty Limits of Parahoric Subgroup of SL(n, Q_p), with Arielle Leitner, Expositiones Mathematicae, Volume 39, Issue 3, September 2021, 500-513, https://doi.org/10.1016/j.exmath.2021.01.001

Although the Chabauty compactification of parahoric subgroups is well studied in a general setting, for the particular case of SL(n, Q_p ) we give a different and more geometric proof of some of the results in Guivarc’h–Rémy 2006 using various Levi decompositions of SL(n,Q_p ).

10. An appendix to Locally compact groups with every isometric action bounded or proper, by Romain Tessera and Alain Valette, Journal of Topology and Analysis, Volume 12, Issue 02, pages 267–292, 2020, https://doi.org/10.1142/S1793525319500547

9. The cone topology on masures, with Bernhard Mühlherr and Guy Rousseau, with an appendix by Auguste Hébert, Advances in

Geometry, Volume 20, Issue 1, Pages 1–28, 2019, https://doi.org/10.1515/advgeom-2019-0020

This preprint improves the essential results in the preprint http://arxiv.org/abs/1504.00526 "Strongly transitive actions on affine ordered hovels" by Corina Ciobotaru and Guy Rousseau.

Masures are generalizations of Bruhat-Tits buildings and the main examples are associated with almost split Kac-Moody groups G over non-Archimedean local fields. In this case, G acts strongly transitively on its corresponding masure Δ as well as on the building at infinity of Δ, which is the twin building associated with G. The aim of this article is twofold: firstly, to introduce and study the cone topology on the twin building at infinity of a masure. It turns out that this topology has various favorable properties that are required in the literature as axioms for a topological twin building. Secondly, by making use of the cone topology, we study strongly transitive actions of a group G on a masure Δ. Under some hypotheses, with respect to the masure and the group action of G, we prove that G acts strongly transitively on Δ if and only if it acts strongly transitively on the twin building at infinity ∂Δ. Along the way a criterion for strong transitivity is given and the existence and good dynamical properties of strongly regular hyperbolic automorphisms of the masure are proven.

8. Infinitely generated Hecke algebras with infinite presentation, Algebras and Representation Theory, 23, 2275–2293 (2020), https://link.springer.com/article/10.1007/s10468-019-09939-8

For a locally compact group G and a compact subgroup K, the corresponding Hecke algebra consists of all continuous compactly supported complex functions on G that are K-bi-invariant. There are many examples of totally disconnected locally compact groups whose Hecke algebras with respect to the maximal compact subgroups are not commutative. One of those is the universal group U(F)^+, when F is primitive but not 2-transitive. For this class of groups we prove that the Hecke algebra with respect to the maximal compact subgroup K is infinitely generated and infinitely presented. This may be relevant for constructing irreducible unitary representations of U(F)^+ whose subspace of K-fixed vectors has dimension at least two, or answering the question whether U(F)^+ is a type I group or not. On the contrary, when F is 2-transitive that Hecke algebra of U(F)^+ is commutative, finitely generated admitting only one generator.

7. A note on type I groups acting on d-regular trees, http://arxiv.org/abs/1506.02950, not for publication.

In this note, we give new examples of type I groups generalizing a previous result of Ol'shanskii. More precisely, we prove that all closed non-compact subgroups of Aut(T_{d}) acting transitively on the vertices and on the boundary of a d-regular tree and satisfying Tits' independence property are type I groups. We claim no originality as we use standard ingredients: the polar decomposition of those groups and the admissibility of all their irreducible unitary representations.

6. Parabolically induced unitary representations of the universal group U(F)^+ are C_0, Mathematica Scandinavica, 125(1), 113–134, https://doi.org/10.7146/math.scand.a-114722

By employing a new strategy we prove that all parabolically induced unitary representations of the Burger-Mozes universal group U(F)^+, with F being primitive, have all their matrix coefficients vanishing at infinity. This generalizes the same well-known result for the universal group U(F)^+, when F is 2-transitive.

5. Positivity of the renormalized volume of almost-Fuchsian hyperbolic 3-manifolds, with Sergiu Moroianu, Proc. Amer. Math. Soc. 144 (2016), 151-159, https://doi.org/10.1090/proc/12682

We prove that the renormalized volume of almost-Fuchsian hyperbolic 3-manifolds is non-negative, with equality only for Fuchsian manifolds.

4. A unified proof of the Howe-Moore property, Journal of Lie Theory 25 (2015), No. 1, 065–089, http://www.heldermann.de/JLT/JLT25/JLT251/jlt25005.htm

We provide a unified proof of all known examples of locally compact groups that enjoy the Howe-Moore property, namely, the vanishing at infinity of all matrix coefficients of the group's unitary representations that are without non-zero invariant vectors. These examples are: the connected, non-compact, simple real Lie groups with finite center, isotropically simple algebraic groups over locally compact non Archimedean fields and closed, topologically simple subgroups of Aut(T) that act 2-transitively on the boundary of T, where T is a bi-regular tree with valence >2 at every vertex.

3. The flat closing problem for buildings, Algebraic and Geometric Topology 14 (2014) 3089–3096, https://msp.org/agt/2014/14-5/agt-v14-n5-p18-s.pdf

Using the notion of a strongly regular hyperbolic automorphism of a locally finite Euclidean building, we prove that any (not necessarily discrete) closed, co-compact subgroup of the type-preserving automorphisms group of a locally finite general non-spherical building contains a compact-by-Z^d subgroup, where d is the dimension of a maximal flat.

2. Gelfand pairs and strong transitivity for Euclidean buildings, with P-E. Caprace, Ergodic Theory and Dynamical Systems, Volume 35, Issue 04, 2015, pp 1056–1078, http://arxiv.org/abs/1304.6210.

Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building X and K be the stabilizer of a special vertex in X. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on X; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse of this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on X, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculiar dynamics on the spherical building at infinity. Other equivalent formulations are also obtained, including the fact that G is strongly transitive on X if and only if it is strongly transitive on the spherical building at infinity.

1. PhD Thesis: Analytic aspects of locally compact groups acting on Euclidean buildings, Université catholique de Louvain, Belgium, Ph.D. supervisor Pierre-Emmanuel Caprace.