ABSTRACT. It is shown that, in a certain statistical sense, in almost every group with rn generators and n relations (with m and n chosen), any subgroup generated by less than m elements (which need not belong to the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked groups.

ABSTRACT. We prove that the commutator is stable in permutations endowed with the Hamming distance, that is, two permutations that almost commute are near two commuting permutations.Our result extends to k-tuples of almost commuting permutations, for any given k, and allows restrictions, for instance, to even permutations.

In this paper we study residual properties of relatively hyperbolic groups. In particular, we show that if a group G is non-elementary and hyperbolic relative to a collection of proper subgroups, then G is SQuniversal.

ABSTRACT. We construct the first example of a coarsely non-amenable (= without Guoliang Yu's property A) metric space with bounded geometry which coarsely embeds into a Hilbert space.

We give first examples of finitely generated groups having an intermediate, with values in (0, 1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group F is equal to 1/2, the Hilbert space compression of Z ≀ Z is between 1/2 and 3/4, and the Hilbert space compression of Z ≀ (Z ≀ Z) is between 0 and 1/2. In general, we find a relationship between the growth of H and the Hilbert space compression of Z ≀ H.