Non-elementary amenable subgroups of automata groups
نویسنده
چکیده
We consider groups of automorphisms of locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. This covers all known examples of groups that are not elementary amenable and act on the trees: groups of intermediate growths and Basilica group, by giving a more straightforward proof. Moreover, we deduce that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.
منابع مشابه
5 F eb 2 01 5 CHAIN CONDITIONS , ELEMENTARY AMENABLE GROUPS , AND DESCRIPTIVE SET THEORY
We first consider three well-known chain conditions in the space of marked groups: the minimal condition on centralizers, the maximal condition on subgroups, and the maximal condition on normal subgroups. For each condition, we produce a characterization in terms of well-founded descriptiveset-theoretic trees. Using these characterizations, we demonstrate that the sets given by these conditions...
متن کاملElementary Amenable Subgroups of R. Thompson's Group F
This paper contributes to the study of the subgroups of Thompson’s group F by constructing a sequence of subgroups of increasing complexity. The group F is an interesting finitely presented group with a pleasant, faithful representation in the group PLo(I) of orientation preserving, piecewise linear, self homeomorphisms of the unit interval. Thus our subgroups also lie in PLo(I). The subgroups ...
متن کاملThe Non-amenability of Schreier Graphs for Infinite Index Quasiconvex Subgroups of Hyperbolic Groups
We show that if H is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group G then the Schreier coset graph for G relative to H is non-amenable.
متن کاملOn a Class of Automata Groups Generalizing Lamplighter Groups
We study automata groups generated by reset automata. Every lamplighter group Z/nZ wr Z can be generated by such an automaton, and in general these automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free. Parabolic subgroups and fractal properties are considered.
متن کاملElementary classes of groups
For a given family of groups B, we define an elementary class E(B) as the smallest class of groups, that contains B and is closed under the passing to subgroups, quotients, and taking extensions and direct limits. The properties of these classes are considered. The proposed theory has close connections with the theory of elementary amenable groups as well as of Kuroš – Černikov classes.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2015