575 research outputs found
A family of uniform lattices acting on a Davis complex with a non-discrete set of covolumes
Let be a Coxeter system with Davis complex . The polyhedral
automorphism group of is a locally compact group under the
compact-open topology. If is a discrete group (as characterised by
Haglund--Paulin), then the set of uniform lattices in is
discrete. Whether the converse is true remains an open problem. Under certain
assumptions on , we show that is non-discrete and
contains rationals (in lowest form) with denominators divisible by arbitrarily
large powers of any prime less than a fixed integer. We explicitly construct
our lattices as fundamental groups of complexes of groups with universal cover
. We conclude with a new proof of an already known analogous result for
regular right-angled buildings
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
Relative outer automorphisms of free groups
Let be a system of free factors of . The group of relative
automorphisms is the group given by the automorphisms of
that restricted to each are conjugations by elements in . The
group of relative outer automorphisms is defined as , where is the normal subgroup of
given by all the inner automorphisms. We define a contractible space
on which acts with finite stabilizers and we compute the
virtual cohomological dimension of this group.Comment: 24 pages, 16 figures, corrected typos, revised argument in section 5,
results unchange
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