Revista Matemática Iberoamericana

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Volume 29, Issue 1, 2013, pp. 75–89
DOI: 10.4171/RMI/713

Published online: 2013-01-14

Relating the Freiheitssatz to the asymptotic behavior of a group

Francisco F. Lasheras[1] and Ranja Roy[2]

(1) Universidad de Sevilla, Spain
(2) New York Institute of Technology, Old Westbury, USA

We are concerned with the implications of the Freiheitssatz property for certain group presentations in terms of proper homotopy invariants of the underlying group, by describing its fundamental pro-group. A finitely presented group G is said to be properly 3-realizable if it is the fundamental group of a finite 2-dimensional CW-complex whose universal cover has the proper homotopy type of a 3-manifold. We show that if an infinite finitely presented group G is given by some special kind of presentation satisfying the Freiheitssatz, then G is semistable at infinity and properly 3-realizable. In particular, this applies to groups given by a staggered presentation.

Keywords: Freiheitssatz, proper homotopy, ends of groups, semistable at infinity, fundamental pro-group, properly 3-realizable, 3-manifold

Lasheras Francisco, Roy Ranja: Relating the Freiheitssatz to the asymptotic behavior of a group. Rev. Mat. Iberoam. 29 (2013), 75-89. doi: 10.4171/RMI/713