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# Documents  Arzhantseva, Goulnara N. | enregistrements trouvés : 6

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## Endomorphisms, train track maps, and fully irreducible monodromies Kapovich, Ilya | CIRM H

Post-edited

Research Talks;Algebra;Dynamical Systems and Ordinary Differential Equations;Geometry;Topology

An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group G depends only on the component of the BNS invariant $\sum \left ( G \right )$ containing the associated homomorphism to the integers. In particular, it follows that if G is the mapping torus of an atoroidal fully irreducible automorphism of a free group and if the union of $\sum \left ( G \right )$ and $\sum \left ( G \right )$ is connected then for every splitting of $G$ as a (f.g. free)-by-(infinite cyclic) group the monodromy is fully irreducible.
This talk is based on joint work with Spencer Dowdall and Christopher Leininger.
An endomorphism of a finitely generated free group naturally descends to an injective endomorphism on the stable quotient. We establish a geometric incarnation of this fact : an expanding irreducible train track map inducing an endomorphism of the fundamental group determines an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully ...

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## Geometric group theory :Geneva and Barcelona conferences#June 20-28 and June 28-July 2 Arzhantseva, Goulnara N. ; Bartholdi, Laurent ; Burillo, Jose ; Ventura, Enric | Birkhäuser 2007

Congrès

- 253 p.
ISBN 978-3-7643-8411-1

Trends in mathematics

Localisation : Colloque 1er étage (GENE)

théorie des groupes # groupes géométriques # algèbre de groupe localement compact

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## Full groups, cost, symmetric groups and IRSS Le Maître, François | CIRM H

Multi angle

Research Talks;Algebra;Algebraic and Complex Geometry

In this talk, we will first review some of the analogies between full groups of measure-preserving equivalence relations and the symmetric group over the integers, which have been used by A. Eisenmann and Y. Glasner to provide interesting examples of invariant random subgroups (IRSs) of the free group.
We will then see how the notion of cost, introduced by G. Levitt, naturally enters this picture. After that, we will explain how a stronger analogy between full groups and the symmetric group over the integers holds in the type III case.
A joint result with A. Kaïchouh which uses this analogy will be presented : full groups of hyperfinite type III equivalence relations have ample generics. This provides a positive answer to a question of A. Kechris and C. Rosendal on the existence of connected Polish group with ample generics.
In this talk, we will first review some of the analogies between full groups of measure-preserving equivalence relations and the symmetric group over the integers, which have been used by A. Eisenmann and Y. Glasner to provide interesting examples of invariant random subgroups (IRSs) of the free group.
We will then see how the notion of cost, introduced by G. Levitt, naturally enters this picture. After that, we will explain how a stronger ...

37A05

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## Growth under random products of automorphisms of a free group Horbez, Camille | CIRM H

Multi angle

Research talks;Algebra

Given a nontrivial conjugacy class $g$ in a free group $F_{N}$, what can we say about the typical growth of g under application of a random product of auto-morphisms of $F_{N}$? I will present a law of large numbers, a central limit theorem and a spectral theorem in this context. Similar results also hold for the growth
of simple closed curves on a closed hyperbolic surface, under application of a random product of mapping classes of the surface. This is partly joint work with François Dahmani.
Given a nontrivial conjugacy class $g$ in a free group $F_{N}$, what can we say about the typical growth of g under application of a random product of auto-morphisms of $F_{N}$? I will present a law of large numbers, a central limit theorem and a spectral theorem in this context. Similar results also hold for the growth
of simple closed curves on a closed hyperbolic surface, under application of a random product of mapping classes of the ...

20F65

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## Natural subgroups of automorphisms Guirardel, Vincent | CIRM H

Multi angle

Research Talks;Algebra;Algebraic and Complex Geometry;Topology

I will survey our common work with Gilbert Levitt about subgroups of automorphisms of hyperbolic and relatively hyperbolic groups : McCool groups, stabilizers of trees, stabilizers of subgroups.

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## Kazhdan projections Drutu, Cornelia | CIRM H

Multi angle

Research talks;Algebra

Kazhdan projections are usually considred objects relevant in operator algebras. In particular, they played a central part in the construction of counter-examples to the Baum-Connes conjecture.
In this talk I shall explain how, in the general setting of a family of representations on Banach spaces, one can reformulate the Kazhdan property "almost invariant implies invariant vectors" in terms of Kazhdan projections, providing also an explicit formula of the latter, using Markov operators associated to a random walk on the group. I will then explain some applications of this new approach.
This is joint work with Piotr Nowak.
Kazhdan projections are usually considred objects relevant in operator algebras. In particular, they played a central part in the construction of counter-examples to the Baum-Connes conjecture.
In this talk I shall explain how, in the general setting of a family of representations on Banach spaces, one can reformulate the Kazhdan property "almost invariant implies invariant vectors" in terms of Kazhdan projections, providing also an explicit ...

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