Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups

2002/11/08 by Elmas Irmak, Irmak, Elmas
Mathematics · #20F38 #57M99 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:20F38 #msc:57M99

paper · pdf · doi:10.48550/arxiv.math/0211139

34 pages, 17 figures

arxiv created 2002/11/08 · arxiv updated 2009/11/30

Abstract

Let S be a closed, connected, orientable surface of genus at least 3, C(S) be the complex of curves on S and Mod_S^* be the extended mapping class group of S. We prove that a simplicial map, λ: C(S) → C(S), preserves nondisjointness (i.e. if α and β are two vertices in C(S) and i(α, β) ≠ 0, then i(λ(α), λ(β)) ≠ 0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of Mod_S^* and f: K → Mod_S^* is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of Mod_S^*.

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