Large random simplicial complexes, II; the fundamental group

We study random simplicial complexes in the multi-parameter model focusing mainly on the properties of the fundamental groups. We describe thresholds for nontrivially and hyperbolicity (in the sense of Gromov) for these groups. Besides, we find domains in the multi-parameter space where these groups have 2-torsion. We also prove that these groups never have odd-prime torsion and their geometric and cohomological dimensions are either 0, 1, 2 or [Formula: see text]. Another result presented in this paper states that aspherical 2-dimensional subcomplexes of random complexes satisfy the Whitehead Conjecture, i.e. all their subcomplexes are also aspherical, with probability tending to one.

ON n-DEFICIENCY OF GROUPS

Let G be a group of type Fn with n ≥ 2, i.e. G admits a finite n-dimensional (n - 1)-aspherical CW-complex X such that π1(X) is isomorphic to G. We introduce the notion of the n-deficiency dn(G) which is a generalized notion of deficiency of G and present various results about the relation between dn(G) and the L2-Betti numbers of G. We also give some partial results about the Eilenberg–Ganea conjecture and the Whitehead conjecture. These results correspond to a generalization of several earlier results [Quart. J. Math. Oxford (2)38 (1987) 35–44; J. Pure Appl. Algebra7 (1976) 241–248; Quart. J. Math. Oxford (2)32 (1981) 45–55; 2-Knots and Their Groups (Cambridge University Press, 1989)].

THE WHITEHEAD ASPHERICITY CONJECTURE AND PERIODIC GROUPS

The Whitehead asphericity conjecture claims that if [Formula: see text] is an aspherical group presentation, then for every [Formula: see text] the subpresentation [Formula: see text] is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introducing almost aspherical presentations (for example, every one-relator group is almost aspherical). It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture. It is also proven that the generalized Whitehead asphericity conjecture holds for Ol'shanskii's presentations of free Burnside groups of large odd exponent, presentations of Tarski monsters and others.

On 2-dimensional aspherical complexes and a problem of J. H. C. Whitehead

In [W] J. H. C. Whitehead posed the following question: ‘Is every subcomplex K of a 2-dimensional aspherical complex L itself aspherical ?’This problem is usually referred to as the ‘Whitehead Conjecture’ though it was only stated in the form of a question. For convenience we treat it also as a conjecture.The Whitehead Conjecture has been proved in special cases: if the subcomplex K has only one 2-cell, and also in the case where π1(K) is either finite, abelian, of free [C] For more partial results see, for example, the introduction of [H1].