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

Details Maps of surface groups to finite groups with no simple loops in the kernel Maps of surface groups to finite groups with no simple loops in the kernel

2000-02-20

arxiv.org/abs/math/0002162v1

Journal of Knot Theory and its Ramifications 9 (2000), 1029-1036

Mathematics

Geometric Topology

Scientific paper

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 $2^{g{2^{2g}}}$.

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