On a Z-Generated Infinite 3-Group: Subgroups and Automorphisms
نویسنده
چکیده
This is the second paper dedicated to a detailed study of the infinite 3-group ?J= (74 r>, 1’3 = rr;3 = 1, y=(y, CT, 2-l) (a recursive definition to be explained later), which was introduced by the author and Narain Gupta in [2] as a group of automorphisms of an infinite regular tree F. As was shown in [4], Y has the remarkable properties of containing a copy of every finite 3-group, and having all its proper quotients finite. The first paper [S] in this detailed study 99 dealt with the presentation problem. Here we study further aspects of its subgroup structure. Our prime objective though is to determine the group of automorphisms of 9. This task represents a large portion of our present paper. The main results are as follows.
منابع مشابه
Automorphisms of a Free Group of Infinite Rank
The problem of classifying the automorphisms of a free group of infinite countable rank is investigated. Quite a reasonable generating set for the group AutF∞ is described. Some new subgroups of this group and structural results for them are presented. The main result says that the group of all automorphisms is generated (modulo the IA-automorphisms) by strings and lower triangular automorphisms.
متن کاملSplittings and automorphisms of relatively hyperbolic groups
We study automorphisms of a relatively hyperbolic group G. When G is oneended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GLn(Z) fixing certain basis elements. When more general parabolic groups are allowed, these ...
متن کاملAutomorphisms of Coxeter Groups of Rank Three
If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.
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 کاملFully Residually Free Groups and Graphs Labeled by Infinite Words
Let F = F (X) be a free group with basis X and Z[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (Z[t], X)-graphs a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich-Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon’s ...
متن کامل