0 Fe b 20 00 Maps of Surface Groups to Finite Groups with No Simple Loops in the Kernel

Let Fg denote the closed orientable surface of genus g. What is the least order finite group, Gg, for which there is a homomorphism ψ : π1(Fg) → Gg so that no nontrivial simple closed curve on Fg represents an element in Ker(ψ)? For the torus it is easily seen that G1 = Z2 × Z2 suffices. We prove here that G2 is a group of order 32 and that an upper bound for the order of Gg is given by g . The previously known upper bound was greater than 2 2g . For any compact surface F there exists a finite group G and a homomorphism ψ : π1(F ) → G such that no nontrivial element in the kernel of ψ can be represented by a simple closed curve. Such a homomorphism is said to have nongeometric kernel. Casson, Gabai, and Skora [5] have each constructed examples of this (see Section 2 for details). The presence of such examples raises a variety of questions relating to the characterization of the finite groups that can occur in this way. This paper addresses the problem of determining the relationship between the genus of F and the order of G. In the case that F is a torus a complete analysis is straightforward. For instance, the natural projection ψ : π1(F ) → H1(F ;Z2) ∼= Z2 ×Z2 has nongeometric kernel. Our first result concerns the genus 2 closed orientable surface, F2. Casson’s construction yields a group of order 2. Skora reduced this order considerably by producing a group of order 2. In Section 3 a group of order 2 = 32, G2, is constructed for which there is a homomorphism ψ2 : π1(F2) → G2 having nongeometric kernel. In Section 4 it is proved that no such example can be constructed using a group of order less than 32. The example in Section 3 is generalized to construct examples for arbitrary genus surfaces in Section 5. The order of the groups constructed is quite small compared to previously constructed examples. As the examples directly generalize the minimal genus 2 example, there is the possibility that they are minimal as well. Acknowledgements Thanks are due to Allan Edmonds for pointing out the proof of Theorem 4.2. The work in Section 5 was motivated by discussions with Dennis Johnson. 1 Notation and Conventions Throughout this paper all surfaces will be closed and orientable. References to basepoints for the fundamental group of a space are omitted. Since the property of being in the kernel of a homomorphism depends only on the conjugacy class of an element, such omissions will not affect the arguments. By a simple loop on a surface we mean an embedding of the circle S . We will say that a homomorphism ψ : π1(F ) → G has geometric kernel if some nontrivial element in the kernel can be represented by a simple loop. Otherwise ψ has nongeometric kernel.