scholarly journals On bireversible Mealy automata and the Burnside problem

2018 ◽  
Vol 707 ◽  
pp. 24-35 ◽  
Author(s):  
Thibault Godin ◽  
Ines Klimann
Keyword(s):  
Burnside Problem ◽  
Mealy Automata

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

scholarly journals ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250052 ◽  
Author(s):  
ALI AKHAVI ◽  
INES KLIMANN ◽  
SYLVAIN LOMBARDY ◽  
JEAN MAIRESSE ◽  
MATTHIEU PICANTIN
Keyword(s):  
Decision Problem ◽  
Decision Procedure ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mealy Automata ◽  
Necessary And Sufficient ◽  
New Criteria

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

Numerical upper bounds on growth of automaton groups

2021 ◽  
pp. 1-33
Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi
Keyword(s):  
Group Theory ◽  
Upper Bounds ◽  
Geometric Invariant ◽  
Finitely Generated ◽  
Grigorchuk Group ◽  
Finitely Generated Group ◽  
Intermediate Growth ◽  
Mealy Automata ◽  
Optimal Bounds ◽  
Algorithmic Procedure

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

scholarly journals Bounds in the restricted Burnside problem

1999 ◽  
Vol 67 (2) ◽  
pp. 261-271 ◽  
Author(s):  
Michael Vaughan-Lee ◽  
E. I. Zel'manov
Keyword(s):  
Burnside Problem ◽  
Current State ◽  
Restricted Burnside Problem

AbstractWe survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of PI-algebras.

Burnside problem for measure preserving groups and for 2-groups of toral homeomorphisms

2013 ◽  
Vol 168 (1) ◽  
pp. 387-396 ◽  
Author(s):  
Nancy Guelman ◽  
Isabelle Liousse
Keyword(s):  
Burnside Problem
Sign in / Sign up

Export Citation Format

Share Document