Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups

By Elmas Irmak. 2002

72 citations recorded by the source corpus.

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 ∗ .

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/159862ef35a8f2631fee9f7b7090d94a00e87cb7

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Citations in the corpus

Curve Complexes and Finite Index Subgroups of Mapping Class Groups (2005, background): “Given a pants decomposition P of S1,n, we say that two curves of P are adjacent with respect to P if there exists a component of S1,n − P containing both curves in its closure; this definition is essential in the work of Irmak [17] [16].”
Injections of Artin groups (2005, unknown): “G R ] 1 7”
Injective simplicial maps of the arc complex (2006, result): “Some similar results on simplicial maps and the applications for orientable surfaces are as follows: Irmak proved that superinjective simplicial maps of the curve complex are induced by homeomorphisms of the surface to classify injective homomorphisms from finite index subgroups of the mapping class”
Injections of mapping class groups (2008, unknown): “G T ]”
Commensurations of the Johnson kernel (2004, methodology): “It is not hard to prove that superinjectivity implies injectivity for all of the above complexes, but this does not follow immediately from the definition (see [13]).”