Generating tuples of free products

Abstract

Grushko's theorem [Mat. Sb. 8 (1940) 169-182] says that any generating tuple (g 1 , . . . , gm) of a free product H * K is Nielsen-equivalent to a tuple (h 1 , . . . , h l , k l+1 , . . . , km) with h i ∈ H and k i ∈ K for all i. The h i and k i are clearly not unique. In this paper we address the extent of this non-uniqueness.

Connected sums of unstabilized Heegaard splittings are unstabilized

“…Both Theorem 10.1 and Theorem 10.2 have algebraic analogues established by R Weidmann [19]. In Weidmann's work, the analogue of an unstabilized Heegaard splitting is a generating set .g 1 ; : : : ; g k / for a group that is not Nielsen equivalent to a generating set of the form .g ; 1/.…”

On the rank of quotients of hyperbolic groups

“…We then give a reformulation of the main result of [7] in that language. For graphs of groups and their fundamental groups we follow the notation of [5]; the discussion about representing generating tuples follows [14].…”