We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.

No abstract

We define relatively quasiconvex subgroups of relatively hyperbolic groups in the sense of Osin and show that such subgroups have expected properties. Also we state several definitions equivalent to the definition of relatively hyperbolic groups in the sense of Osin.

Let g and n be integers at least 2, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian. YOSHIKATA KIDA AND SAEKO YAMAGATAHBPs for S, into CP(S) is induced by an element of Mod * (S). Combining the above theorem with [22, Theorem 7.13 (i)] and applying argument in [17, Section 3], we obtain the following: Corollary 1.2. Let S be the surface in Theorem 1.1. Then for any finite index subgroup Γ of P (S) and any injective homomorphism f : Γ → P (S), there exists a unique element γ ∈ Mod * (S) with f (x) = γxγ −1 for any x ∈ Γ. In particular, Γ is co-Hopfian.Parallel results are proved for the subgroup P s (S) of P (S) and the subcomplex CP s (S) of CP(S), called the complex of HBCs and separating HBPs for S, that are precisely defined in Section 2.2. Let us summarize the results on them.Theorem 1.3. Let S be the surface in Theorem 1.1. Then any superinjective map from CP s (S) into itself is induced by an element of Mod * (S).Corollary 1.4. Let S be the surface in Theorem 1.1. Then for any finite index subgroup Λ of P s (S) and any injective homomorphism h : Λ → P s (S), there exists a unique element λ ∈ Mod * (S) with h(y) = λyλ −1 for any y ∈ Λ. In particular, Λ is co-Hopfian.This corollary is obtained by combining the last theorem with [22, Theorem 7.13 (ii)]. If the genus of a surface S is equal to 0, then we have the equalities P (S) = P s (S) = PMod(S) and CP(S) = CP s (S) = C(S). If the genus of S is equal to 1, then we have the equalities P (S) = I(S) and P s (S) = K(S), where I(S) is the Torelli group for S and K(S) is the Johnson kernel for S. Moreover, CP(S) and CP s (S) are equal to the Torelli complex for S and the complex of separating curves for S, respectively. We refer to [19] for a definition of these groups and complexes. It therefore follows from [3] and [20] that if S is a surface with the genus less than 2 and the Euler characteristic less than −2, then the same conclusions for S as in the above theorems hold. The co-Hopfian property of the braid groups on the disk, which are central extensions of the mapping class groups of holed spheres, is discussed in [2] and [4].For a positive integer n and a manifold M , we define B n (M ) as the braid group of n strands on M , i.e., the fundamental group of the space of non-ordered distinct n points in M . We also define P B n (M ) as the pure braid group of n strands on M , i.e., the fundamental group of the space of ordered distinct n points in M . The group P B n (M ) is naturally identified with a subgroup of B n (M ) of index n!. We refer to [5] and [27] for fundamental facts on these groups.Let S be a surface of genus at least 2 with p boundary components. The kernel of the homomorphism from Mod(S) onto Mod(S) associated with the inclusion of S intoS is then identified with B p

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3.Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n ≥ 2. The group G is acting on the universal cover of M , which is isometric to H n C . It is a CAT(−1) space of dimension 2n. The geometric dimension of G is 2n − 1. We show that G does not act on any proper CAT(0) space of dimension 2n − 1 properly by isometries.We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag-Solitar groups.