We present a proof that cohomology classes in H 2,2 (S 1 × S 2 ) of Hodge isometries ψ : H 2 (S 1 , Q) → H 2 (S 2 , Q) between any two projective complex K3 surfaces S 1 and S 2 are polynomials in Chern classes of coherent sheaves. This proves a conjecture of Shafarevich [27].

As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a generic chain of twistor lines. Furthermore, we show that twistor lines are holomorphic submanifolds of Compl, of degree 2 in the Plücker embedding of Compl.

Consider the rank n free group F n with basis X. Bogopol skiȋ conjectured in [1, Problem 15.35] that each element w ∈ F n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F n , as well as the separability by a subgroup of index at most |w| 2 + 2 in general.

In this work we generalize the classical notion of a twistor line in the period domain of compact complex tori studied in [1]. We introduce two new types of lines, which are non-compact analytic curves in the period domain. We study the analytic properties of the compactifications of the curves, preservation of cohomology classes of type (1,1) along them and prove the twistor path connectivity of the period domain by the curves of one of the new types.

Granted the three integers n ≥ 2, r, and R, consider all ordered tuples of r elements of length at most R in the free group F n . Calculate the number of those tuples that generate in F n a rank r subgroup and divide it by the number of all tuples under study. As R → ∞, the limit of the ratio is known to exist and equal 1 (see [1]). We give a simple proof of this result.Denote the cardinality of an arbitrary finite set A by #A.Take a finitely generated group G. Call the least number of generators for G the rank of G and denote it by rk(G). Fix some tuple of generators for G. Then we can equip G with the standard metric d(u, v) = |uv −1 |, where u and v are the vertices of the left Cayley graph of G on which G acts by the right multiplication, and |w| stands for the length of w.TheDenote by G r the direct product of r copies of G and by 1, the tuple (1, . . . , 1) ∈ G r . On G r there is the induced metric D: given u = (u 1 , . . . , u r ) ∈ G r and v = (v 1 , . . . , v r ) ∈ G r , we haveObserve that the radius R ball G r (R) ⊂ G r is the direct product r i=1 G(R) of balls. Calculate the cardinality of G r (R) in the case that G = F n is a rank n free group. In F n there is precisely one element of length zero (the identity element), and there are precisely 2n(2n − 1) k−1 elements of length k for k ≥ 1, and so #F n (R) = 1 + 2n + 2n(2n − 1) + · · · + 2n(2n − 1) R−1