AUXILIARY PROBLEMS

These are problems motivated by some of the OUTSTANDING PROBLEMS. We have started this section in the hope that problems here, some of which are actually interesting in their own right, might generate some hints for attacking the relevant outstanding problems.

*(AUX1) (A. Myasnikov) (cf. Problem (O9))   Let m, n be positive integers, and H and K nontrivial subgroups of a free group such that rank(H) = n and rank(K) = m. Which numbers between 1 and (n-1)(m-1) can be realized as rank(H \cap K) - 1? In particular, can (n-1)(m-1) - 1 be realized? Background

(AUX2) (A. Myasnikov) (cf. Problem (O9))   Is there an algorithm which for every pair of positive integers m, n determines whether or not the Hanna Neumann conjecture holds for all subgroups of ranks m, n in a free group F_2 ?

(AUX3) (A.Borovik, A. Lubotzky, A. Myasnikov) (cf. Problem (O1)) For a group G, by d(G) we denote the minimal number of normal generators of G, and by N_k(G), k \ge d(G), the set of all k-tuples of elements of G which generate G as a normal subgroup. We say that a group G satisfies the generalized Andrews-Curtis conjecture if for any k \ge max(d(G), 2), two tuples U, V \in N_k(G) are AC-equivalent in G if and only if their images are AC-equivalent in the abelianization of G.

(a) Find a group G which does not satisfy the generalized Andrews-Curtis conjecture.

(b) Does Grigorchuk's group satisfy the generalized Andrews-Curtis conjecture?

(AUX4) (A. Casson) (cf. Problem (O7)) Suppose \phi is an onto homomorphism from F_{p+q} to F_p \times F_q. Is it true that Ker(\phi) is the normal closure of precisely pq elements?

(AUX5) (S.V.Ivanov) (cf. Problem (O1))
(a) In addition to "elementary" operations on a set Y described in Problem (O1)(a), consider the following operation: replace some y_i by a free generator of the group F if this does not change the normal closure of Y. Is it true that every n-element set Y whose normal closure coincides with the whole F=F_n can be reduced to X by a sequence of operations in Problem (O1)(a) together with this new operation?

(b) (weaker question) Similar question, but the ONLY operation is: replace some y_i by an element of the group F if this does not change the normal closure of Y. This problem is open if the cardinality of Y is greater than 2, and has the affirmative answer if it is equal to 2.

See [S.V.Ivanov, On balanced presentations of the trivial group, Invent. Math. 165 (2006), 525--549] for relevant discussions.