Maps of surface groups to finite groups with no simple loops in the kernel
Abstract
Let $F_g$ denote the closed orientable surface of genus $g$. What is the least order finite group, $G_g$, for which there is a homomorphism $\psi$ from $\pi_1(F_g)$ to $G_g$ so that no nontrivial simple closed curve on $F_g$ represents an element in Ker($\psi$)? For the torus it is easily seen that $G_1 = Z_2 \times Z_2$ suffices. We prove here that $G_2$ is a group of order 32 and that an upper bound for the order of $G_g$ is given by $g^{2g +1}$. The previously known upper bound was greater than $2^{g{2^{2g}}}$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 February 2000
 arXiv:
 arXiv:math/0002162
 Bibcode:
 2000math......2162L
 Keywords:

 Mathematics  Geometric Topology;
 57M05
 EPrint:
 Journal of Knot Theory and its Ramifications 9 (2000), 10291036