Topology and Group Theory Seminar: January 22, 2025
Speaker: Mike Mihalik (Vanderbilt)
Title: Stallings’ group is simply connected at infinity
Abstract: For n ≥ 2, let Bn denote the kernel of the homomorphism from the direct product of n-copies of the free group on two generators to the group of integers which sends all generators to the generator 1. The groups Bn are called the Bieri-Stallings groups and Bn is of type Fn-1 but not Fn. Classical results can be used to show that Bn is (n-3)-connected at infinity for n ≥ 3. Since Bn is not type Fn it is not (n-1)-connected at infinity. Stallings’ proved that B2 is finitely generated but not finitely presented. We conjecture that for n ≥ 2, Bn is (n-2)-connected at infinity. For n = 2, this would mean that B2 is 1-ended and for n = 3 that B3 (typically called Stallings’ group) is simply connected at infinity. We verify the conjecture for n = 2 and n = 3. Our main result is the case n = 3: Stalling’s group is simply connected at infinity.