Groups
Topological, Combinatorial and Arithmetic Aspects
Edited by T. W. Müller
Publisher: Cambridge University Press
Print Publication Year: 2004
Online Publication Date:November 2009
Online ISBN:9780511550706
Paperback ISBN:9780521542876
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511550706.014
Subjects: Algebra, Logic, categories and Sets
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Introduction
Automorphisms of regular 1-rooted trees of finite valency have been the subject of vigorous investigations in recent years as a source of remarkable groups which reflect the recursiveness of these trees (see [S1], [G2]). It is not surprising that the recursiveness could be interpreted in terms of automata. Indeed, the automorphisms of the tree have a natural interpretation as input-output automata where the states, finite or infinite in number, are themselves automorphisms of the tree. On the other hand input-output automata having the same input and output alphabets can be seen as endomorphisms of a 1-rooted tree indexed by finite sequences from this alphabet. It is to be noted that the set of automorphisms having a finite number of states and thus corresponding to finite automata, form an enumerable group called the group of finite-state automorphisms. The calculation of the product of two automorphisms of the tree involve calculating products between their states which are not necessarily elements of the group generated by the two automorphisms. In order to remain within the same domain of calculation we have defined a group G as state-closed provided the states of its elements are also elements of G [S2]. Among the outstanding examples of state-closed groups are the classes of self-reproducing (fractal-like) groups constructed in [G1, GS, BSV] which are actually generated by automorphisms with finite number of states, or equivalently, generated by finite automata.
pp. i-iv
pp. v-vi
List of authors and participants : Read PDF
pp. vii-x
pp. xi-xvi
1 - Reductive groups as metric spaces : Read PDF
pp. 1-20
2 - Finiteness properties of groups acting on twin buildings : Read PDF
pp. 21-26
3 - Higher finiteness properties of S-arithmetic groups in the function field case I : Read PDF
pp. 27-42
4 - Controlled topology and group actions : Read PDF
pp. 43-63
5 - Finiteness properties of soluble S-arithmetic groups – A survey : Read PDF
pp. 64-92
6 - Topology in permutation groups : Read PDF
pp. 93-105
7 - Euler characteristics of discrete groups : Read PDF
pp. 106-254
8 - Intersections of Magnus subgroups of one-relator groups : Read PDF
pp. 255-296
9 - A minimality property of certain branch groups : Read PDF
pp. 297-305
10 - Lattices with non-integral character : Read PDF
pp. 306-317
11 - Some applications of probability in group theory : Read PDF
pp. 318-326
12 - Parity patterns in Hecke groups and Fermat primes : Read PDF
pp. 327-374
13 - Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms : Read PDF
pp. 375-404
14 - The mapping class group of the twice punctured torus : Read PDF
pp. 405-486
15 - Kac–Moody groups: split and relative theories. Lattices : Read PDF
pp. 487-541
16 - On the finite images of infinite groups : Read PDF
pp. 542-563
17 - Pseudo-finite generalized triangle groups : Read PDF
pp. 564-587