1,509 research outputs found
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 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 on flag
manifolds associated to and Finsler compactifications of associated
symmetric spaces . Find domains of proper discontinuity and use them to
construct natural bordifications and compactifications of the locally symmetric
spaces .Comment: 77 page
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.Comment: 88 page
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.Comment: 93 page
Two constructions with parabolic geometries
This is an expanded version of a series of lectures delivered at the 25th
Winter School ``Geometry and Physics'' in Srni.
After a short introduction to Cartan geometries and parabolic geometries, we
give a detailed description of the equivalence between parabolic geometries and
underlying geometric structures.
The second part of the paper is devoted to constructions which relate
parabolic geometries of different type. First we discuss the construction of
correspondence spaces and twistor spaces, which is related to nested parabolic
subgroups in the same semisimple Lie group. An example related to twistor
theory for Grassmannian structures and the geometry of second order ODE's is
discussed in detail.
In the last part, we discuss analogs of the Fefferman construction, which
relate geometries corresponding different semisimple Lie groups
Topology of random simplicial complexes: a survey
This expository article is based on a lecture from the Stanford Symposium on
Algebraic Topology: Application and New Directions, held in honor of Gunnar
Carlsson, Ralph Cohen, and Ib Madsen.Comment: After revisions, now 21 pages, 5 figure
Dynamics on flag manifolds: domains of proper discontinuity and cocompactness
For noncompact semisimple Lie groups we study the dynamics of the actions
of their discrete subgroups on the associated partial flag manifolds
. 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 -action on various domains of proper
discontinuity, in particular on domains in the full flag manifold . We
show in the regular case (of -Anosov subgroups) that the latter domains are
always nonempty if if has (locally) at least one noncompact simple factor
not of the type or .Comment: 65 page
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