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

الأصل · EN

Let S be a closed, connected, orientable surface of genus at least 3, C(S) be the complex of curves on S and Modₛ* 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 Modₛ* and f: K → Modₛ* is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of Modₛ*.