Domains of discontinuity of Lorentzian affine group actions
We prove nonemptyness of domains of proper discontinuity of Anosov groups of affine Lorentzian transformations.
math.GR↗
arXiv subjects
Bernhard Leeb
Publications and source records attributed to Bernhard Leeb.
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↗
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↗
A combination theorem for Anosov subgroups
We prove an analogue of Klein combination theorem for Anosov subgroups by using a local-to-global principle for Morse quasigeodesics.
math.GR↗
Discrete isometry groups of symmetric spaces
This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $Γ<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete subgroups $Γ$ on flag manifolds associated to $G$ and Finsler compactifications of associated symmetric spaces $X=G/K$. Find domains of proper discontinuity and use them to construct natural bordifications and compactifications of the locally symmetric spaces $X/Γ$.
math.GR↗
Anosov subgroups: Dynamical and geometric characterizations
We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of rank one Lie groups to higher rank. Our main goal is to give several new equivalent characterizations for this important class of discrete subgroups. Our characterizations capture "rank one behavior" of Anosov subgroups and are direct generalizations of rank one equivalents to convex cocompactness. Along the way, we considerably simplify the original definition, avoiding the geodesic flow. We also show that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group.
math.GR↗
Dynamics on flag manifolds: domains of proper discontinuity and cocompactness
For noncompact semisimple Lie groups $G$ we study the dynamics of the actions of their discrete subgroups $Γ<G$ on the associated partial flag manifolds $G/P$. Our study is based on the observation that they exhibit also in higher rank a certain form of convergence type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the $Γ$-action on various domains of proper discontinuity, in particular on domains in the full flag manifold $G/B$. We show in the regular case (of $B$-Anosov subgroups) that the latter domains are always nonempty if if $G$ has (locally) at least one noncompact simple factor not of the type $A_1, B_2$ or $G_2$.
math.MG↗
Finsler bordifications of symmetric and certain locally symmetric spaces
We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces $X/Γ$ for arbitrary discrete subgroups $Γ< G$. These bordifications result from attaching $Γ$-quotients of suitable domains of proper discontinuity at infinity. We further prove that such bordifications are compactifications in the case of Anosov subgroups. We show, conversely, that Anosov subgroups are characterized by the existence of such compactifications among uniformly regular subgroups. Along the way, we give a positive answer, in the torsion free case, to a question of Haïssinsky and Tukia on convergence groups regarding the cocompactness of their actions on the domains of discontinuity.
math.DG↗
Some recent results on Anosov representations
In this note we give an overview of some of our recent work on Anosov representations of discrete groups into higher rank semisimple Lie groups.
math.GR↗
A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings
We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furthermore that they must be word hyperbolic. We introduced this class of discrete subgroups in our earlier paper, in the context of symmetric spaces, where various equivalent geometric and dynamical characterizations of word hyperbolic Morse subgroups were established, including the Anosov subgroup property.
math.GR↗
Morse actions of discrete groups on symmetric space
We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. A substantial part of the paper is devoted to the coarse geometry of these discrete subgroups. A key concept which emerges from our analysis is that of Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse property for quasigeodesics in Gromov hyperbolic spaces. It leads to the notion of Morse actions of word hyperbolic groups on symmetric spaces,i.e. actions for which the orbit maps are Morse quasiisometric embeddings, and thus provides a coarse geometric characterization for the class of subgroups considered in this paper. A basic result is a local-to-global principle for Morse quasigeodesics 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.GR↗
A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry
We study geodesically complete and locally compact Hadamard spaces X whose Tits boundary is a connected irreducible spherical building. We show that X is symmetric iff complete geodesics in X do not branch and a Euclidean building otherwise. Furthermore, every boundary equivalence (cone topology homeomorphism preserving the Tits metric) between two such spaces is induced by a homothety. As an application, we can extend the Mostow and Prasad rigidity theorems to compact singular (orbi)spaces of nonpositive curvature which are homotopy equivalent to a quotient of a symmetric space or Euclidean building by a cocompact group of isometries.
math.MG↗
Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds
We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [32].
math.GT↗
Induced quasi-actions: a remark
In this note we observe that the notion of an induced representation has an analog for quasi-actions. We then use induced quasi-actions to refine some earlier rigidity results for product spaces.
math.GR↗
Convex functions on symmetric spaces, side lengths of polygons and stability inequalities for weighted configurations at infinity
In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond to points in the Euclidean Weyl chamber. We can hence assign vector-valued side-lengths to segments. Our main result is a system of homogeneous linear inequalities describing the restrictions on the side -lengths of closed polygons. The inequalities are based on the mod 2 Schubert calculus in the real Grassmannians G/P for maximal parabolic subgroups P.
math.DG↗
Polygons in buildings and their refined side lengths
As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued side lengths of polygons in (thick) Euclidean buildings. The main result is that the set of possible side lengths of n-gons depends only on the associated spherical Coxeter complex. We show moreover that it coincides with the space of vector valued weights of semistable weighted configurations on the Tits boundary. The side lengths of polygons in symmetric spaces of noncompact type are studied in the related paper [KLM1]. Applications of the geometric results in both papers to algebraic group theory are given in [KLM3].
math.MG↗
The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra
In this paper we apply our results on the geometry of polygons in Cartan subspaces, symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the computation of the structure constants of the Hecke and representation rings associated with a split reductive algebraic group over Q and its complex Langlands' dual. We give a new proof of the ``Saturation Conjecture'' for GL(m) as a consequence of our solution of the corresponding ``saturation problem'' for the Hecke structure constants for all split reductive algebraic groups over Q.
math.RT↗
Rigidity of invariant convex sets in symmetric spaces
The main result implies that a proper convex subset of an irreducible higher rank symmetric space cannot have Zariski dense stabilizer.
math.DG↗