86 research outputs found
Usos de proprio en italiano y sus traducciones al español: la bilógica como clave para el análisis contrastivoy el estudio de las interferencias
Extensive amenability and an application to interval exchanges
Extensive amenability is a property of group actions which has recently been
used as a tool to prove amenability of groups. We study this property and prove
that it is preserved under a very general construction of semidirect products.
As an application, we establish the amenability of all subgroups of the group
IET of interval exchange transformations that have angular components of
rational rank~.
In addition, we obtain a reformulation of extensive amenability in terms of
inverted orbits and use it to present a purely probabilistic proof that
recurrent actions are extensively amenable. Finally, we study the triviality of
the Poisson boundary for random walks on IET and show that there are subgroups
admitting no finitely supported measure with trivial boundary.Comment: 28 page
A commutator lemma for confined subgroups and applications to groups acting on rooted trees
A subgroup of a group is confined if the -orbit of under
conjugation is bounded away from the trivial subgroup in the space
of subgroups of . We prove a commutator lemma for
confined subgroups. For groups of homeomorphisms, this provides the exact
analogue for confined subgroups (hence in particular for URSs) of the classical
commutator lemma for normal subgroups: if is a group of homeomorphisms of a
Hausdorff space and is a confined subgroup of , then contains
the derived subgroup of the rigid stabilizer of some open subset of . We
apply this commutator lemma to the study of URSs and actions on compact spaces
of groups acting on rooted trees. We prove a theorem describing the structure
of URSs of weakly branch groups and of their non-topologically free minimal
actions. Among the applications of these results, we show: 1) if is a
finitely generated branch group, the -action on has the
smallest possible orbital growth among all faithful -actions; 2) if is a
finitely generated branch group, then every embedding from into a group of
homeomorphisms of strongly bounded type (e.g. a bounded automaton group) must
be spatially realized; 3) if is a finitely generated weakly branch group,
then does not embed into the group IET of interval exchange
transformations.Comment: 48 pages. v1->v2: minor revisio
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