Event Type:

Number Theory Seminar

Date/Time:

Tuesday, November 11, 2014 - 16:00 to 17:00

Location:

Graf 307

Local Speaker:

Abstract:

This talk concerns groups $J_n(m,k)$ defined by presentations $(t,y: t^n, y^{m-k}*t^3*y^k*t^2)$ where $n=4$ or $6$. These presentations arose in the study of asphericity for relative presentations of the form $(H,y:r)$ where $r$ is a word of length four in the free product $H \ast \<y\>$. The $J$-groups represent limiting cases where aspherical presentations transition to non-aspherical ones. In joint work with Gerald Williams (Essex) we prove that if $m \neq 0$ and $k$ are relatively prime, then $J_n(m,k)$ is finite with abelian derived subgroup, usually cyclic. However if in addition $m-2k \equiv 0 \mod n$, then the derived subgroup $J_n(m,k)'$ is a direct product of two cyclic groups with equal orders. All Mersenne primes (known or unknown) and other conjecturally infinite families of primes occur as orders of these factors.