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.
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Francisco F. Lasheras, Ranja Roy, Relating the Freiheitssatz to the asymptotic behavior of a group. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 75–89DOI 10.4171/RMI/713