Download e-book for iPad: Algebraic Generalizations of Discrete Groups: A Path to by Benjamin Fine

By Benjamin Fine

ISBN-10: 0824703197

ISBN-13: 9780824703196

A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic examine of Fuchsian teams to the extra common context of one-relator items and comparable workforce theoretical issues. It offers a self-contained account of yes typical generalizations of discrete teams.

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Surv. 356 (2004), No. 2, 65–104. [70] F. O. Wagner, Relational structures and dimensions, pp. 153–180 in Automorphisms of First-Order Structures (ed. R. W. Kaye and H. D. Macpherson), Oxford University Press, Oxford, 1994. [71] H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964. [72] H. Wielandt, Unendliche Permutationsgruppen, Lecture Notes, Universit¨ at T¨ ubingen, 1959. [English translation by P. Bruyns included in Wielandt’s collected works: H. edu Contents 1 Self-similar objects 38 2 Actions on rooted trees 41 3 Self-similar groups 44 4 Iterated monodromy groups 50 5 Branch groups 56 6 Growth of groups 66 7 Amenability 72 8 Schreier graphs related to self-similar groups 81 Introduction The idea of self-similarity is one of the most basic and fruitful ideas in mathematics of all times and populations.

4n 3 What happens for primitive groups? The first result asserts the existence of a wide gap, between constant and exponential growth. /p(n) for some polynomial p. 148 . .. 324 . .. No counterexamples are known to the conjecture that both parts hold with c = 2. The growth rates of the sequences (fn ) and (Fn ) appear to have a great deal of structure; little is known, but examples suggest some conjectures. One of the most challenging conjectures is to prove that various ‘obvious’ limits, such as lim(log fn )/(log n) (for polynomial growth), lim(log log fn )/(log n) (for fractional exponential growth), or lim(log fn )/n (for exponential growth), actually exist.

Self-similarity and branching in group theory Grigorchuk, Suni c 44 Definition 3 The homomorphism ϕu is called the projection of G at u. The image of ϕu is denoted by Gu and called the upper companion group of G at u. The map ψn : StG (Ln ) → Aut(T ) u∈Ln given by ψn (g) = (ϕu (g))u∈Ln = (gu )u∈Ln is a homomorphism. We usually omit the index in ψn when n = 1. In the case of G = Aut(T ) the maps ϕu : StAut(T ) (u) → Aut(T ) and ψn : StAut(T ) (Ln ) → u∈Ln Aut(T ) are isomorphisms, for any word u over X and any n ≥ 0.

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Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products by Benjamin Fine


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