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

Elmas Irmak

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arXiv · math/0311407

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

Elmas Irmak

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Abstract

Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R.

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BibTeX RIS

Elmas Irmak. 2003-12-29. Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups II. https://arxiv.org/abs/math/0311407

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