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Corina Ciobotaru

Publications and source records attributed to Corina Ciobotaru.

At least 19 recordsLinked to original sources
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

Covers of Bruhat-Tits trees

Let $G$ be a locally compact group and let $\widetilde{G}$ be a central extension that splits over a maximal compact subgroup $K$ of $G$. We derive an explicit cocycle that lifts the natural action of $G$ on the homogeneous space $G/K$ to an action of $\widetilde{G}$. As an application, for a non-Archimedean local field $F$, we construct a connected locally finite tree on which the metaplectic covers of $\operatorname{GL}_2(F)$ act by automorphisms, providing a geometric analog of the Bruhat--Tits tree of $\operatorname{GL}_2(F)$. Furthermore, under suitable transitivity assumptions, we prove that $(\widetilde{G},\widetilde{K})$ is a Gelfand pair. Finally, we describe the associated parabolic and contraction subgroups with respect to $\widetilde{G}$ from the perspective of the geometry of the constructed tree.

math.RT↗
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

Chabauty limits of fixed point groups of $p$-adic involutions

We study Chabauty limits of the fixed-point group of $k$-points $H_k$ associated with an involutive $k$-automorphism $θ$ of a connected linear reductive group $G$ defined over a non-Archimedean local field $k$ of characteristic zero. Leveraging the geometry of the Bruhat--Tits building, the structure of $(θ,k)$-split tori, and the $K\mathcal{B}_kH_k$ decomposition of $G_k$, we establish that any nontrivial Chabauty limit $L$ of $H_k$ is $G_k$-conjugate to a subgroup of $$U_{σ_+}(k) \rtimes (Ker(α)^0 \cdot (H_k \cap M_{σ_{\pm}})) \leq P_{σ_+}(k),$$ where $α$ is a projection map arising from a Levi factor $M_{σ_{\pm}}$ of a parabolic subgroup $P_{σ_+} \subset G$, and $Ker(α)^0$ denotes the subgroup of elliptic elements in the kernel of $α$. Our analysis distinguishes between elliptic and hyperbolic elements and constructs explicit unipotent elements in the limit group $L$ using the Moufang property of $G_k$. Furthermore, we show that $L$ acts transitively on the set of ideal simplices opposite to $σ_+$. These results yield a detailed description of the Chabauty compactification of $H_k$, and provide new insights into its interaction with the non-Archimedean geometry of $G_k$.

math.RT↗
arXiv2024

Dynamics of strongly I-regular hyperbolic elements on affine buildings

The first goal of this article is to investigate a refinement of previously-introduced strongly regular hyperbolic automorphisms of locally finite thick Euclidean buildings $Δ$ of finite Coxeter system $(W,S)$. The new ones are defined for each proper subset $I \subsetneq S$ and called strongly $I$-regular hyperbolic automorphisms of $Δ$. Generalizing previous results, we show that such elements exist in any group $G$ acting cocompactly and by automorphisms on $Δ$. Although the dynamics of strongly $I$-regular hyperbolic elements $γ$ on the spherical building $\partial_\infty Δ$ of $Δ$ is much more complicated than for the strongly regular ones, the $\lim\limits_{n\to \infty} γ^{n}(ξ)$ still exists in $\partial_\infty Δ$ for ideal points $ξ\in \partial_\infty Δ$ that satisfy certain assumptions. An important role in this business is played by the cone topology on $Δ\cup \partial_\infty Δ$ and the projection of specific residues of $\partial_\infty Δ$ on the ideal boundary of $Min(γ)$. All the above research is performed in order to achieve the second, and main, goal of the article. Namely, we prove that for closed groups $G$ with a type-preserving and strongly transitive action by automorphisms on $Δ$, the Chabauty limits of certain closed subgroups of $G$ contain as a normal subgroup the entire unipotent radical of concrete parabolic subgroups of $G$.

math.GR↗
arXiv2023

Symmetry breaking for $\operatorname{PGL}(2)$ over non-archimedean local fields

For a quadratic extension $\mathbb{E}/\mathbb{F}$ of non-archimedean local fields we construct explicit holomorphic families of intertwining operators between principal series representations of $\operatorname{PGL}(2,\mathbb{E})$ and $\operatorname{PGL}(2,\mathbb{F})$, also referred to as symmetry breaking operators. These families are given in terms of their distribution kernels which can be viewed as distributions on $\mathbb{E}$ depending holomorphically on the principal series parameters. For all such parameters we determine the support of these distributions, and we study their mapping properties. This leads to a classification of all intertwining operators between principal series representations, not necessarily irreducible. As an application, we show that every Steinberg representation of $\operatorname{PGL}(2,\mathbb{E})$ contains a Steinberg representation of $\operatorname{PGL}(2,\mathbb{F})$ as a direct summand of Hilbert spaces.

math.RT↗
arXiv2023

Chabauty limits of groups of involutions in $SL(2,F)$ for local fields

We classify Chabauty limits of groups fixed by various (abstract) involutions over $SL(2,F)$, where $F$ is a finite field-extension of $\mathbb{Q}_p$, with $p\neq 2$. To do so, we first classify abstract involutions over $SL(2,F)$ with $F$ a quadratic extension of $\mathbb{Q}_p$, and prove $p$-adic polar decompositions with respect to various subgroups of $p$-adic $SL_2$. Then we classify Chabauty limits of: $SL(2, F) \subset SL(2,E)$ where $E$ is a quadratic extension of $F$, of $SL(2,\mathbb{R}) \subset SL(2,\mathbb{C})$, and of $H_θ\subset SL(2,F)$, where $H_θ$ is the fixed point group of an $F$-involution $θ$ over $SL(2,F)$.

math.GR↗
arXiv2023

Polyhedral compactifications, I

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 polyhedral 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.

math.MG↗
arXiv2023

On wonderful compactifications of $SL(2,F)$ for non-Archimedean local fields $F$

We compute the wonderful compactification of symmetric varieties of $SL(2,F)$, where $F$ is a finite field-extension of $Q_p$ with $p\neq 2$, that comes from either an abstract or $F$-involutions of $SL(2,F)$. For each of those wonderful compactifications we find the $SL(2,F)$-stabilizers of the accumulation points of the corresponding symmetric varieties and compare them to the Chabauty limits found in Ciobotaru--Leitner 2022.

math.GR↗
arXiv2021

Strong transitivity, Moufang's condition and the Howe--Moore property

Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $Δ$ of dimension $\geq 2$ that acts strongly transitively on $Δ$ is Moufang. If moreover $Δ$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\G(k)^+$ of the $k$-rational points $\G(k)$ of the isotropic simple algebraic group $\G$ over a non-Archimedean local field $k$ associated with $Δ$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $Δ$, and obtain that any topologically simple, closed, strongly transitive and type-preserving subgroup of $\Aut(Δ)$ has the Howe--Moore property. This proof is different than the strategy used so far in the literature and does not relay on the polar decomposition $KA^+K$, where $K$ is a maximal compact subgroup, and the important fact that $A^+$ is an abelian maximal sub-semi-group.

math.GR↗
arXiv2021

(Non)-escape of mass and equidistribution for horospherical actions on trees

Let $G$ be a large group acting on a biregular tree $T$ and $Γ\leq G$ 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 there is no escape of mass and use this to prove that, in fact, dense orbits equidistribute to the Haar measure on $G/Γ$. 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 $Γ\backslash T$ of the uniform distributions on large spheres in the tree $T$ converge to a natural probability measure on $Γ\backslash T$. Finally, we apply this equidistribution result to a lattice point counting problem to obtain counting asymptotics with exponential error term.

math.DS↗
arXiv2021

The universal group of Burger--Mozes and the Howe--Moore property

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.

math.GR↗
arXiv2020

Geometrical and statistical properties of M-estimates of scatter on Grassmann manifolds

We consider data from the Grassmann manifold $G(m,r)$ of all vector subspaces of dimension $r$ of $\mathbb{R}^m$, and focus on the Grassmannian statistical model which is of common use in signal processing and statistics. Canonical Grassmannian distributions $\mathbb{G}_Σ$ on $G(m,r)$ are indexed by parameters $Σ$ from the manifold $\mathcal{M}= Pos_{sym}^{1}(m)$ of positive definite symmetric matrices of determinant $1$. Robust M-estimates of scatter (GE) for general probability measures $\mathcal{P}$ on $G(m,r)$ are studied. Such estimators are defined to be the maximizers of the Grassmannian log-likelihood $-\ell_{\mathcal{P}}(Σ)$ as function of $Σ$. One of the novel features of this work is a strong use of the fact that $\mathcal{M}$ is a CAT(0) space with known visual boundary at infinity $\partial \mathcal{M}$. We also recall that the sample space $G(m,r)$ is a part of $\partial \mathcal{M}$, show the distributions $\mathbb{G}_Σ$ are $SL(m,\mathbb{R})$--quasi-invariant, and that $\ell_{\mathcal{P}}(Σ)$ is a weighted Busemann function. Let $\mathcal{P}_n =(δ_{U_1}+\cdots+δ_{U_n})/n$ be the empirical probability measure for $n$-samples of random i.i.d. subspaces $U_i\in G(m,r)$ of common distribution $\mathcal{P}$, whose support spans $\mathbb{R}^m$. For $Σ_n$ and $Σ_{\mathcal{P}}$ the GEs of $\mathcal{P}_n$ and $\mathcal{P}$, we show the almost sure convergence of $Σ_n$ towards $Σ$ as $n\to\infty$ using methods from geometry, and provide a central limit theorem for the rescaled process $C_n = \frac{m}{tr(Σ_{\mathcal{P}}^{-1} Σ_n)}g^{-1} Σ_n g^{-1}$, where $Σ=gg$ with $g\in SL(m,\mathbb{R})$ the unique symmetric positive-definite square root of $Σ$.

math.ST↗
arXiv2019

Measure rigidity for horospherical subgroups of groups acting on trees

We investigate analogues of some of the classical results in homogeneous dynamics in non-linear setting. Let $G$ be a closed subgroup of the group of automorphisms of a biregular tree and $Γ<G$ a discrete subgroup. For a large class of groups $G$ we give a classification of probability measures on $G/Γ$ invariant under horospherical subgroups. When $Γ$ is a cocompact lattice, we prove unique ergodicity of the horospherical action. We prove Hedlund's theorem for geometrically finite quotients. Finally, we study equidistribution of large compact orbits.

math.DS↗
arXiv2018

Chabauty Limits of Subgroups of $SL(n, \mathbb{Q}_p)$

We study the Chabauty compactification of two families of closed subgroups of $SL(n,\mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,\mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Levi decompositions of $SL(n,\mathbb{Q}_p)$. Let $C$ be the subgroup of diagonal matrices in $SL(n, \mathbb{Q}_p)$. The second family is the set of all $SL(n,\mathbb{Q}_p)$-conjugates of $C$. We give a classification of the Chabauty limits of conjugates of $C$ using the action of $SL(n,\mathbb{Q}_p)$ on its associated Bruhat--Tits building and compute all of the limits for $n\leq 4$ (up to conjugacy). In contrast, for $n\geq 7$ we prove there are infinitely many $SL(n,\mathbb{Q}_p)$-nonconjugate Chabauty limits of conjugates of $C$. Along the way we construct an explicit homeomorphism between the Chabauty compactification in $\mathfrak{sl}(n, \mathbb{Q}_p)$ of $SL(n,\mathbb{Q}_p)$-conjugates of the $p$-adic Lie algebra of $C$ and the Chabauty compactification of $SL(n,\mathbb{Q}_p)$-conjugates of $C$.

math.GT↗
arXiv2018

The cone topology on masures

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 $\partial$$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.

math.GR↗
arXiv2018

Mean field repulsive Kuramoto models: Phase locking and spatial signs

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 ow 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 id2, the variance-covariance matrix of a random vector that is uniformly distributed on the unit circle.

nlin.AO↗