Free subgroups in groups acting on rooted trees
نویسندگان
چکیده
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 groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki. Mathematics Subject Classification (2010). 20E08, 20E05.
منابع مشابه
Dimension and randomness in groups acting on rooted trees by Miklós Abért ∗ and
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 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 کاملGidi Amir - The Liouville property for groups acting on rooted trees
This is a summary on verbal subgroups in the groups T(∞, K) and UT(∞, K) of infinite dimensional triangular and unitriangular matrices over a field K, |K| > 2. The characterization of the lattices of verbal subgroups in these groups is presented, and the width of verbal subgroups generated by non-commutators, powers and (generalized) Engel words is given [2, 3, 4]. The groups T(∞, K) and UT(∞, ...
متن کاملIndecomposable Groups Acting on Hyperbolic Spaces
We obtain a number of finiteness results for groups acting on hy-perbolic spaces and R-trees. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups. We also prove an acylindrical accessibility theorem for groups acting on R-trees.
متن کاملLimit groups and groups acting freely on R n-trees.
We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R n-trees. We first prove that Sela's limit groups do have a free action on an R n-tree. We then prove that a finitely generated group having a free action on an R n-tree can be obtained from free abelian groups and surface groups by a finite sequence of free products and amalgamations over cyclic ...
متن کامل