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

الأصل · EN

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 from π₁(Fg) to 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 g²ᵍ ⁺¹. The previously known upper bound was greater than 2g2²ᵍ.