2004/07/16 by Elmas Irmak, Irmak, Elmas · 3 citations
Mathematics · #20F38 #57M99 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #math.GT #msc:20F38 #msc:57M99
paper · pdf · doi:10.48550/arxiv.math/0407285
24 pages, 13 figures; The result about automorphism group of complex of nonseparating curves has been extended to compact, connected, orientable surfaces of genus at least two
openalex publication_date 2004/07/16 · arxiv created 2004/08/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a compact, connected, orientable surface of genus g, ModR^* be the extended mapping class group of R, C(R) be the complex of curves on R, and N(R) be the complex of nonseparating curves on R. We prove that if g ≥ 2 and R has at most g-1 boundary components, then a simplicial map λ: N(R) → N(R) is superinjective if and only if it is induced by a homeomorphism of R. We prove that if g ≥ 2 and R is not a closed surface of genus two then Aut(N(R))= ModR^*, and if R is a closed surface of genus two then Aut(N(R))= ModR ^* /C(ModR^*). We also prove that if g=2 and R has at most one boundary component, then a simplicial map λ: C(R) → C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary we prove some new results about injective homomorphisms from finite index subgroups to ModR^*. The last two results complete the author's previous results to connected orientable surfaces of genus at least two.