A commutator lemma for confined subgroups and applications to groups acting on rooted trees
نویسندگان
چکیده
A subgroup H H of a group G"> G encoding="application/x-tex">G is confined if the -orbit under conjugation bounded away from trivial in space S u b left-parenthesis upper G right-parenthesis"> Sub ( stretchy="false">) encoding="application/x-tex">\operatorname {Sub}(G) subgroups . We prove commutator lemma for subgroups. For groups homeomorphisms, this provides exact analogue (hence particular uniformly recurrent (URSs)) classical normal subgroups: homeomorphisms Hausdorff X"> X encoding="application/x-tex">X and , then contains derived rigid stabilizer some open subset apply setting acting on rooted trees. theorem describing structure URSs weakly branch their non-topologically free minimal actions. Among applications these results, we show: (1) finitely generated group, -action alttext="partial-differential T"> ∂ T encoding="application/x-tex">\partial T has smallest possible growth among all faithful -actions; (2) every embedding into strongly type (e.g. automaton group) must be spatially realized; (3) does not embed IET interval exchange transformations.
منابع مشابه
Free subgroups in groups acting on rooted trees
We show that if a group G acting faithfully on a rooted tree T has a free subgroup, then either there exists a point w of the boundary @T and a free subgroup of G with trivial stabilizer ofw, or there existsw 2 @T and a free subgroup ofG fixingw and acting faithfully on arbitrarily small neighborhoods of w. This can be used to prove the absence of free subgroups for different known classes of g...
متن کاملDimension and randomness in groups acting on rooted trees
We explore the structure of the p-adic automorphism group Γ of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free...
متن کاملThe Automorphism Tower of Groups Acting on Rooted Trees
The group of isometries Aut(Tn) of a rooted n-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in Aut(Tn). This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group Γ̈ studied by N. Gupta and the second author. In this paper, we pursue t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2023
ISSN: ['2330-0000']
DOI: https://doi.org/10.1090/tran/8965