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The finitary Andrews-Curtis conjecture. (English) Zbl 1114.20011
Bartholdi, Laurent (ed.) et al., Infinite groups: geometric, combinatorial and dynamical aspects. Based on the international conference on group theory: geometric, combinatorial and dynamical aspects of infinite groups, Gaeta, Italy, June 1–6, 2003. Basel: Birkhäuser (ISBN 3-7643-7446-2/hbk). Progress in Mathematics 248, 15-30 (2005).
Summary: The well known Andrews-Curtis conjecture [J. J. Andrews and M. L. Curtis, Proc. Am. Math. Soc. 16, 192-195 (1965; Zbl 0131.38301)] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked by A. V. Borovik, E. I. Khukhro and A. G. Myasnikov [in Int. J. Algebra Comput. 13, No. 4, 415-436 (2003; Zbl 1053.20023)] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [loc. cit.].
For the entire collection see [Zbl 1083.20500].

MSC:
20E05 Free nonabelian groups
20F05 Generators, relations, and presentations of groups
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