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Bulletin of the London Mathematical Society Advance Access originally published online on April 8, 2008
Bulletin of the London Mathematical Society 2008 40(3):375-383; doi:10.1112/blms/bdn017
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© 2008 London Mathematical Society

Closed subgroups of free profinite monoids are projective profinite groups

John Rhodes

Department of Mathematics
University of California at Berkeley
Berkeley, CA 94720
USA
rhodes@math.berkeley.edu

Benjamin Steinberg

School of Mathematics and Statistics
Carleton University
1125 Colonel By Drive
Ottawa
Ontario
Canada K1S 5B6

Received 30 November 2006. Revision received 16 October 2007.

We prove that the class of closed subgroups of free profinite monoids is precisely the class of projective profinite groups. In particular, the profinite groups associated to minimal symbolic dynamical systems by Almeida are projective. Our result answers a question raised by Lubotzky during the lecture of Almeida at the Fields Workshop on Profinite Groups and Applications, Carleton University, August 2005. We also prove that any finite subsemigroup of a free profinite monoid consists of idempotents.

2000 Mathematics Subject Classification 20F20, 20M07.

The second author was supported in part by NSERC.


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