108 research outputs found
The Polish topology of the isometry group of the infinite dimensional hyperbolic space
We consider the isometry group of the infinite dimensional separable
hyperbolic space with its Polish topology. This topology is given by the
pointwise convergence. For non-locally compact Polish groups, some striking
phenomena like automatic continuity or extreme amenability may happen. Our
leading idea is to compare this topological group with usual Lie groups on one
side and with non-Archimedean infinite dimensional groups like
, the group of all permutations of a countable set on the
other side. Our main results are
Automatic continuity (any homomorphism to a separable group is continuous),
minimality of the Polish topology, identification of its universal Furstenberg
boundary as the closed unit ball of a separable Hilbert space with its weak
topology, identification of its universal minimal flow as the completion of
some suspension of the action of the additive group of the reals on its
universal minimal flow.
All along the text, we lead a parallel study with the sibling group of
isometries of a separable Hilbert space.Comment: After a first version of this paper, Todor Tsankov asked if the
topology is minimal. A positive answer has been added to this second versio
Topological properties of Wazewski dendrite groups
Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite
countable set of branch points of the dendrite and thus have a nice Polish
topology. In this paper, we study them in the light of this Polish topology.
The group of the universal Wa\.zewski dendrite is more
characteristic than the others because it is the unique one with a dense
conjugacy class. For this group , we show some of its topological
properties like existence of a comeager conjugacy class, the Steinhaus
property, automatic continuity and the small index subgroup property. Moreover,
we identify the universal minimal flow of . This allows us to prove
that point-stabilizers in are amenable and to describe the universal
Furstenberg boundary of .Comment: Slight modifications about the expositio
Superrigidity In Infinite Dimension And Finite Rank Via Harmonic Maps
We prove geometric superrigidity for actions of cocompact lattices in
semisimple Lie groups of higher rank on infinite dimensional Riemannian
manifolds of nonpositive curvature and finite telescopic dimension
Amenable Invariant Random Subgroups
We show that an amenable Invariant Random Subgroup of a locally compact
second countable group lives in the amenable radical. This answers a question
raised in the introduction of the paper "Kesten's Theorem for Invariant Random
Subgroup" by Abert, Glasner and Virag. We also consider, in the opposite
direction, property (T), and prove a similar statement for this property. The
Appendix by Phillip Wesolek proves that the set of amenable subgroups is a
Borel subset in the Chabauty topology.Comment: We added an Appendix by Phillip Wesole
Almost algebraic actions of algebraic groups and applications to algebraic representations
Let G be an algebraic group over a complete separable valued field k. We
discuss the dynamics of the G-action on spaces of probability measures on
algebraic G-varieties. We show that the stabilizers of measures are almost
algebraic and the orbits are separated by open invariant sets. We discuss
various applications, including existence results for algebraic representations
of amenable ergodic actions. The latter provides an essential technical step in
the recent generalization of Margulis-Zimmer super-rigidity phenomenon due to
Bader and Furman.Comment: Correction of a small mistake in Proposition 5.
A group with Property (T) acting on the circle
We exhibit a topological group with property (T) acting non-elementarily
and continuously on the circle. This group is an uncountable totally
disconnected closed subgroup of . It has
a large unitary dual since it separates points. It comes from homeomorphisms of
dendrites and a kaleidoscopic construction. Alternatively, it can be seen as
the group of elements preserving some specific geodesic lamination of the
hyperbolic disk.
We also prove that this action is unique up to conjugation and that it can't
be smoothened in any way. Finally, we determine the universal minimal flow of
the group
Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator
We associate to any Riemannian symmetric space (of finite or infinite
dimension) a L-algebra, under the assumption that the curvature operator
has a fixed sign. L-algebras are Lie algebras with a pleasant Hilbert space
structure. The L-algebra that we construct is a complete local isomorphism
invariant and allows us to classify Riemannian symmetric spaces with fixed-sign
curvature operator. The case of nonpositive curvature is emphasized.Comment: A para\^itre aux annales de l'institut Fourie
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