The co-Hopfian property of surface braid groups. (Q2859561)

Given a surface \(S\), associated to \(S\) there is the so called complex of curves for \(S\), denoted by \(\mathcal C(S)\). Many results are known about the automorphism group of this complex. Variants of this complex of curves where introduced in several places. One of the variants, denoted by \(\mathcal{CP}(S)\), was introduced by the authors [in J. Math. Soc. Japan 63, No. 4, 1391-1435 (2011; Zbl 1245.20041)]. The main stream of the work under review is to study superinjective self-maps of \(\mathcal{CP}(S)\).NEWLINENEWLINE They show: Theorem 1.1. Let \(S\) be a connected, compact and orientable surface such that both the genus and the number of boundary components of \(S\) are at least \(2\). Then any superinjective map from \(\mathcal{CP}(S)\) into itself is induced by an element of \(\mathrm{Mod}^*(S)\).NEWLINENEWLINE Pure braid groups are related with the mapping class of the surface \(S\). Denote by \(P(S)\) the pure braid group on \(p\) (\(p\) is the number of components of the boundary of \(S\)) strings of the surface \(\overline S\), where \(\overline S\) is the closed surface obtained from \(S\) by attaching disks to all boundary components of \(S\).NEWLINENEWLINE As a corollary of the theorem above they show: Corollary 1.2. Let \(S\) be the surface in Theorem 1.1. Then for any finite index subgroup \(\Gamma\) of \(P(S)\) and any injective homomorphism \(f\colon\Gamma\to P_S(S)\), there exists a unique element \(\gamma\in\mathrm{Mod}^*(S)\) with \(f(x)=\gamma x\gamma^{-1}\) for any \(x\in\Gamma\). In particular \(\Gamma\) is co-Hopfian.NEWLINENEWLINE Then they show a consequence of the above corollary which is: Corollary 1.5. Let \(n\) be an integer at least 2. Let \(M\) be a connected, closed and orientable surface of genus at least \(2\). Then any finite index subgroup of \(B_n(M)\) is co-Hopfian.NEWLINENEWLINE A subcomplex, denoted by \(\mathbb{CP}_s(S)\) and it has to do with separated curves, of \(\mathbb{C}(S)\) is defined as well as a suitable subgroup \(P_s(S)\) of \(P(S)\).NEWLINENEWLINE Then they show similar results for this new setting. Namely Theorem 1.3. Let \(S\) be the surface in Theorem 1.1. Then any superinjective map from \(\mathbb{CP}_s(S)\) to itself is induced by an element of \(\mathrm{Mod}^*(S)\).NEWLINENEWLINE Corollary 1.4. Let \(S\) be the surface in Theorem 1.1. Then for any finite index subgroup \(\Lambda\) of \(P_s(S)\) and any injective homomorphism \(h\colon\Lambda\to P_S(S)\), there exists a unique element \(\lambda\in\mathrm{Mod}^*(S)\) with \(h(y)=\lambda y\lambda^{-1}\) for any \(y\in\Lambda\). In particular, \(\Lambda\) is co-Hopfian.NEWLINENEWLINE The manuscript presents a quite detailed description of preliminaries and previous results. Also it contains a good set of figures which help to follow the proofs, which contain many details.