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Journal für die reine und angewandte Mathematik

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Random quotients of the modular group are rigid and essentially incompressible

Ilya Kapovich1 / Paul E. Schupp2

1Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA. e-mail:

2Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA. e-mail:

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2009, Issue 628, Pages 91–119, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2009.019, January 2009

Publication History

Received:
2006-04-19
Revised:
2007-11-16
Published Online:
2009-01-21

Abstract

We show that for any positive integer m ≧ 1, m-relator quotients of the modular group M = PSL(2,ℤ) generically satisfy a very strong Mostow-type isomorphism rigidity. We also prove that such quotients are generically “essentially incompressible”. By this we mean that their “absolute T-invariant”, measuring the smallest size of any possible finite presentation of the group, is bounded below by a function which is almost linear in terms of the length of the given presentation. We compute the precise asymptotics of the number Im(n) of isomorphism types of m-relator quotients of M where all the defining relators are cyclically reduced words of length n in M. We obtain other algebraic results and show that such quotients are complete, Hopfian, co-Hopfian, one-ended, word-hyperbolic groups.

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