Automorphisms of profinite mapping class groups

2020/11/30 by Boggi, Marco
#14G32 #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2011.15075

Abstract

For S=S_g,n a closed orientable differentiable surface of genus g from which n points have been removed, such that χ(S)=2-2g-n<0, let PΓ(S) be the pure mapping class group of S and P\widehatΓ(S) and P\checkΓ(S) be, respectively, its profinite and its congruence completions. The latter can be identified with the image of the natural representation P\widehatΓ(S)\toOut(\widehatπ₁(S)), where \widehatπ₁(S) is the profinite completion of the fundamental group of the surface S. Let Out^\mathbbI₀(P\widehatΓ(S)) and Out^\mathbbI₀(P\checkΓ(S)) be the groups of outer automorphisms which preserve the conjugacy class of a procyclic subgroup generated by a nonseparating Dehn twist and let \widehatGT be the profinite Grothendieck-Teichmüller group. We then prove that, for χ(S)

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