Groups acting on trees with finitely generated free subgroups of finite index

2013 ◽  
Vol 7 ◽  
pp. 5321-5326
Author(s):  
R. M. S. Mahmood
Keyword(s):  
Finite Index ◽  
Finitely Generated ◽  
Groups Acting On Trees ◽  
Free Subgroups

scholarly journals Statistics of subgroups of the modular group

2021 ◽  
pp. 1-61
Author(s):  
Frédérique Bassino ◽  
Cyril Nicaud ◽  
Pascal Weil
Keyword(s):  
Finite Index ◽  
Modular Group ◽  
Large Deviation ◽  
Free Subgroup ◽  
Index Formula ◽  
Index Subgroup ◽  
Finitely Generated ◽  
Finite Index Subgroup ◽  
Free Subgroups ◽  
Finite Index Subgroups

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

scholarly journals Finitely generated subgroups of amalgamated free products and HNN groups

1976 ◽  
Vol 22 (3) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel E. Cohen
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Subnormal Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Hnn Extension ◽  
Groups Acting On Trees

AbstractThe theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Groups in which every Finitely Generated Subgroup is almost a Free Factor

1979 ◽  
Vol 31 (6) ◽  
pp. 1329-1338 ◽  
Author(s):  
A. M. Brunner ◽  
R. G. Burns
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Products

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

On Certain Finitely Generated Subgroups of Groups Which Split

2003 ◽  
Vol 46 (1) ◽  
pp. 122-129 ◽  
Author(s):  
Myoungho Moon
Keyword(s):  
Positive Integer ◽  
Free Product ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Product With Amalgamation

AbstractDefine a group G to be in the class 𝒮 if for any finitely generated subgroup K of G having the property that there is a positive integer n such that gn ∈ K for all g ∈ G, K has finite index in G. We show that a free product with amalgamation A *CB and an HNN group A *C belong to 𝒮, if C is in 𝒮 and every subgroup of C is finitely generated.

scholarly journals DISTORTION IN FREE NILPOTENT GROUPS

2010 ◽  
Vol 20 (05) ◽  
pp. 661-669 ◽  
Author(s):  
TARA C. DAVIS
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.

scholarly journals Locally graded groups with all subgroups normal-by-finite

1996 ◽  
Vol 60 (2) ◽  
pp. 222-227 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Locally Finite ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Open Question ◽  
Locally Graded Groups

AbstractIn a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

Sign in / Sign up

Export Citation Format

Share Document