Searcharxiv⌕ Search

arXiv subjects

Michael Kapovich

Publications and source records attributed to Michael Kapovich.

At least 19 recordsLinked to original sources
arXiv2026

Equidistribution of currents under Anosov group actions

We study the behavior of currents on flag-manifolds under actions of Anosov subgroups of complex semisimple Lie groups G. Given a current T of bidimension (k, k) on the full flag-manifold F = G/B, we average it under the group action via a construction analogous to the construction of Patterson-Sullivan measures. We show that under certain genericity conditions (in the case of currents of integration over subvarieties), the limiting current is a Gibbs current, i.e. is given by integration (with respect to a Gibbs measure) over the flag-limit set of suitable multiples of currents of integration along Schubert varieties based at the limit points of the group. We prove that the same equidistribution result (but without any genericity assumptions) for currents defined by smooth forms on F. We also prove a form of Axiom A property for the suspension flow of the group action on F.

math.DS↗
arXiv2024

Klein-Maskit combination theorem for Anosov subgroups: Amalgams

The classical Klein-Maskit combination theorems provide sufficient conditions to construct new Kleinian groups using old ones. There are two distinct but closely related combination theorems: The first deals with amalgamated free products, whereas the second deals with HNN extensions. This article gives analogs of both combination theorems for Anosov subgroups.

math.GR↗
arXiv2023

A note on laminations with symmetric leaves

We prove that (apart from dimension $n=4$), each Riemannian solenoidal lamination with transitive homeomorphism group and leaves isometric to a symmetric space $X$ of noncompact type, is homeomorphic to the inverse limit of the system of finite covers of a compact locally-symmetric $n$-manifold.

math.GT↗
arXiv2023

Klein-Maskit combination theorem for Anosov subgroups: Free products

We prove a generalization of the classical Klein-Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if $Γ_A$ and $Γ_B$ are Anosov subgroups of a semisimple Lie group $G$ of noncompact type, then under suitable topological assumptions, the group generated by $Γ_A$ and $Γ_B$ in $G$ is again Anosov, and is naturally isomorphic to the free product $Γ_A*Γ_B$. Such a generalization was conjectured in our previous article with Bernhard Leeb (arXiv:1805.07374).

math.GR↗
arXiv2023

Relativizing characterizations of Anosov subgroups, I

We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically hyperbolic to relatively hyperbolic subgroups. We prove implications and equivalences between the various relativizations.

math.GR↗
arXiv2023

Morse actions of discrete groups on symmetric spaces: Local-to-global principle

Our main result is a local-to-global principle for Morse quasigeodesics, maps and actions. As an application of our techniques we show algorithmic recognizability of Morse actions and construct Morse ``Schottky subgroups'' of higher rank semisimple Lie groups via arguments not based on Tits' ping-pong. Our argument is purely geometric and proceeds by constructing equivariant Morse quasiisometric embeddings of trees into higher rank symmetric spaces.

math.DG↗
arXiv2022

Structural stability of meandering-hyperbolic group actions

In his 1985 paper Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of "meandering hyperbolicity" for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain non-hyperbolic groups, such as actions of "uniform lattices" in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.

math.GR↗
arXiv2022

Effective bounds for Vinberg's algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

math.GT↗
arXiv2022

Patterson-Sullivan theory for Anosov subgroups

We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups.

math.GR↗
arXiv2022

Trees of hyperbolic spaces

We give an alternative proof of the Bestvina--Feighn combination theorem for trees hyperbolic spaces and describe uniform quasigeodesics in such spaces. As one of the applications, we prove the existence of Cannon-Thurston maps for inclusion maps of total spaces of subtrees of hyperbolic spaces.

math.GR↗
arXiv2021

On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups

We develop the notion of a Kleinian Sphere Packing, a generalization of "crystallographic" (Apollonian-like) sphere packings defined by Kontorovich-Nakamura [KN19]. Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do "superintegral" such. We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from Q-arithmetic lattices of simplest type. The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles pi/m for finitely many m. We settle two questions from [KN19]: (i) that the Arithmeticity Theorem is in general false over number fields, and (ii) that integral packings only arise from non-uniform lattices.

math.NT↗
arXiv2020

A note on complex-hyperbolic Kleinian groups

Let $Γ$ be a discrete group of isometries acting on the complex hyperbolic $n$-space $\mathbb{H}^n_\mathbb{C}$. In this note, we prove that if $Γ$ is convex-cocompact, torsion-free, and the critical exponent $δ(Γ)$ is strictly lesser than $2$, then the complex manifold $\mathbb{H}^n_\mathbb{C}/Γ$ is Stein. We also discuss several related conjectures.

math.GR↗