Relating the Freiheitssatz to the asymptotic behavior of a group

  • Francisco F. Lasheras

    Universidad de Sevilla, Spain
  • Ranja Roy

    New York Institute of Technology, Old Westbury, USA

Abstract

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.

Cite this article

Francisco F. Lasheras, Ranja Roy, Relating the Freiheitssatz to the asymptotic behavior of a group. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 75–89

DOI 10.4171/RMI/713