Injections of Artin groups. (Q2468433)

This paper concerns finite-index-subgroups of the extended mapping class group of a sphere with at least 5 punctures. The main theorem shows that if \(\rho\) is an injection from such a subgroup into the mapping class group, then it is conjugation by a unique element of the subgroup. Several consequences of this theorem are given, in particular the observation that all such subgroups are co-Hopfian (i.e. their injective endomorphisms are all isomorphisms). The theorem is proved using techniques of \textit{N. V. Ivanov} [Topology and geometry, Rohlin Semin. 1984-1986, Lect. Notes Math. 1346, 199-270 (1988; Zbl 0657.57004)], translating problems into questions about the curve complex.