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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The geometry of profinite graphs with applications to free groups and finite monoids

Authors: K. Auinger and B. Steinberg
Journal: Trans. Amer. Math. Soc. 356 (2004), 805-851
MSC (2000): Primary 20E18, 20E08, 20M07, 20M18
Published electronically: August 21, 2003
MathSciNet review: 2022720
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Abstract: We initiate the study of the class of profinite graphs $\Gamma$ defined by the following geometric property: for any two vertices $v$ and $w$ of $\Gamma$, there is a (unique) smallest connected profinite subgraph of $\Gamma$ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition.

We define a pseudovariety of groups $\mathbf{H}$ to be arboreous if all finitely generated free pro- $\mathbf{H}$ groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties $\mathbf{H}$, a pro- $\mathbf{H}$ analog of the Ribes and Zalesski{\u{\i}}\kern.15em product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions $\mathbf{H}$ to the much studied pseudovariety equation $\mathbf{J}\ast\mathbf{H}= \mathbf{J}\mathrel{{\mbox{\textcircled{\petite m}}}}\mathbf{H}$.

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Additional Information

K. Auinger
Affiliation: Institut für Mathematik, Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

B. Steinberg
Affiliation: School of Mathematics and Statistics, Carleton University, Herzberg Laboratories, Ottawa, Ontario, Canada K1S 5B6

Received by editor(s): September 28, 2002
Received by editor(s) in revised form: March 13, 2003
Published electronically: August 21, 2003
Additional Notes: The authors gratefully acknowledge support from INTAS project 99–1224 Combinatorial and geometric theory of groups and semigroups and its applications to computer science. The second author was supported in part by NSF-NATO postdoctoral fellowship DGE-9972697, as well as by Praxis XXI scholarship BPD 16306 98, FCT through the Centro de Matemática da Universidade do Porto, and by the FCT and POCTI approved project POCTI/32817/MAT/2000 in participation with the European Community Fund FEDER. He was at the University of Porto when this work was performed.
Article copyright: © Copyright 2003 American Mathematical Society