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Critical constants for recurrence of random walks on G -spaces

Anna Erschler (2005)

Annales de l’institut Fourier

We introduce the notion of a critical constant c r t for recurrence of random walks on G -spaces. For a subgroup H of a finitely generated group G the critical constant is an asymptotic invariant of the quotient G -space G / H . We show that for any infinite G -space c r t ≥ 1 / 2 . We say that G / H is very small if c r t < 1 . For a normal subgroup H the quotient space G / H is very small if and only if it is finite. However, we give examples of infinite very small G -spaces. We show also that critical constants for recurrence can be used...

Equivalence relations induced by some locally compact groups of homeomorphisms of 2 ℕ

B. Majcher-Iwanow (2005)

Colloquium Mathematicae

Let T be a locally finite rooted tree and B(T) be the boundary space of T. We study locally compact subgroups of the group TH(B(T)) = ⟨Iso(T),V⟩ generated by the group Iso(T) of all isometries of B(T) and the group V of Richard Thompson. We describe orbit equivalence relations arising from actions of these groups on B(T).

Folner sets of alternate directed groups

Jérémie Brieussel (2014)

Annales de l’institut Fourier

An explicit family of Folner sets is constructed for some directed groups acting on a rooted tree of sublogarithmic valency by alternate permutations. In the case of bounded valency, these groups were known to be amenable by probabilistic methods. The present construction provides a new and independent proof of amenability, using neither random walks, nor word length.

Généralisation d'un théorème de Haagerup

Ferdaous Kellil, Guy Rousseau (2005)

Studia Mathematica

Let G be a group of automorphisms of a tree X (with set of vertices S) and H a kernel on S × S invariant under the action of G. We want to give an estimate of the l r -operator norm (1 ≤ r ≤ 2) of the operator associated to H in terms of a norm for H. This was obtained by U. Haagerup when G is the free group acting simply transitively on a homogeneous tree. Our result is valid when X is a locally finite tree and one of the orbits of G is the set of vertices at even distance from a given vertex; a technical...

Générateurs indépendants pour les systèmes d'isométries de dimension un

Damien Gaboriau (1997)

Annales de l'institut Fourier

Un système fini d’isométries partielles de R est dit à générateurs indépendants si les composés non triviaux fixent au plus un point. On décrit un procédé simple et naturel pour obtenir des générateurs indépendants, sans modifier les orbites, pour tout système sans composante minimale homogène : en prenant la restriction de chaque générateur à un certain sous-intervalle de son domaine. Un système avec une composante minimale homogène ne possède pas de générateurs indépendants.

Groups of given intermediate word growth

Laurent Bartholdi, Anna Erschler (2014)

Annales de l’institut Fourier

We show that there exists a finitely generated group of growth ∼ f for all functions f : ℝ + → ℝ + satisfying f ( 2 R ) ≤ f ( R ) 2 ≤ f ( η + R ) for all R large enough and η + ≈ 2 . 4675 the positive root of X 3 - X 2 - 2 X - 4 . Set α - = log 2 / log η + ≈ 0 . 7674 ; then all functions that grow uniformly faster than exp ( R α - ) are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth ∼ exp ( R α ) for a dense set of α ∈ [ α - , 1 ] .

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