Published on *Department of Mathematics* (https://math.yale.edu)

October 23, 2018 - 4:00pm

DL 431

The rank of a group is the minimal cardinality of a generating set. While simple to define, this quantity is notoriously difficult to calculate, and often uncomputable, even for well behaved groups. In this talk I will explain general conditions that may be used to show many hyperbolic group extensions have rank equal to the rank of the kernel plus the rank of the quotient and, further, that any minimal generating set is Nielsen equivalent to one in a standard form. This builds on work of Souto on fundamental groups of fibered hyperbolic three manifolds and of Scott-Swarup who in this setting proved that infinite-index finitely generated subgroups of the fiber are quasi-convex in the ambient group. As an application, we prove that if $g_1,\dotsc,g_k$ are independent, atoroidal, fully irreducible outer automorphisms of the free group $F_n$, then there is a power $m$ so that the subgroup generated by $f_1^m,\dotsc,f_k^m$ gives rise to a hyperbolic extension of $F_n$ of rank $n+k$. Joint work with Sam Taylor.