Maps of surface groups to finite groups with no simple loops in the kernel (Q2706860)

Consider a homomorphism \(\psi: \pi_1(F_g)\to G\) of groups, where \(F_g\) denotes the closed orientable surface of genus \(g\). An early unsuccessful approach to the 3-dimensional Poincaré conjecture sought to determine when some nontrivial element of the kernel of a homomorphism \(\psi\) must be represented by a simple closed curve on \(F_g\). One discovered that, even when \(G\) is finite, the homomorphism \(\psi\) may be nongeometric in the sense that no nontrivial element of the kernel of \(\psi\) is so represented. The author seeks to find the minimal possible order of a group \(G_g\) for which there is a nongeometric homomorphism \(\psi:\pi_1(F_g)\to G_g\). This size was known to be 4 for \(g=1\). The author succeeds in showing that the minimal order is 32 for \(g=2\). He lowers the known upper bounds substantially for all \(g\geq 2\) by giving an explicit construction. The proof that this construction yields minimal order for \(g=2\) employs substantial theorems about group actions on surfaces by J. Nielsen and A. Edmonds.