Commensurators of surface braid groups

Yamagata, Saeko

doi:10.2969/jmsj/06341391

Cited by 7 publications

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“…Irmak-Ivanov-McCarthy proved that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three in [11]. Kida and Yamagata also proved several results about superinjective simplicial maps in [22], [23], [24], and as applications they gave a description of any injective homomorphism from a finite index subgroup of the pure braid group with n strands on a closed orientable surface of genus g into the pure braid group. They proved that 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.…”

Section: Introductionmentioning

confidence: 99%

On simplicial maps of the complexes of curves of nonorientable surfaces

Irmak

2014

Algebr. Geom. Topol.

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Section: Introductionmentioning

confidence: 99%

On simplicial maps of the complexes of curves of nonorientable surfaces

Irmak

2014

Algebr. Geom. Topol.

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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.

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confidence: 77%

The Co-Hopfian Property of Surface Braid Groups

Yamagata

2013

J. Knot Theory Ramifications

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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

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“…This is a version of the complex of curves of S that was introduced in [56] to compute virtual automorphisms of P (S), inspired by [41,43]. Finally, to conclude our result, we will use the results from [56,57] that describe all simplicial automorphisms of CP(S) as well as the structure of a certain injection from a subcomplex of CP(S) into CP(S). In the case of I(S) and K(S), we will use similar results from [6,7,17,54,58] which are applicable to the corresponding versions of the complex of curves.…”

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confidence: 99%

“…The following simplicial complex CP(S) was introduced in [56], inspired by the work of Irmak-Ivanov-McCarthy [41], to compute virtual automorphisms of P (S).…”

Section: Complexes For Surface Braid Groupsmentioning

confidence: 99%

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OEandW* superrigidity results for actions by surface braid groups

Chifan

2015

Proc. London Math. Soc.

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A. We show that several important normal subgroups Γ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action Γ X is stably OE-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings.Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preserving action of almost any surface braid group is stably W * -superrigid, i.e., it can be completely reconstructed from its von Neumann algebra.

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Commensurators of surface braid groups

Abstract: 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.

Cited by 7 publications

References 20 publications

On simplicial maps of the complexes of curves of nonorientable surfaces

On simplicial maps of the complexes of curves of nonorientable surfaces

The Co-Hopfian Property of Surface Braid Groups

OEandW* superrigidity results for actions by surface braid groups

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