Yoshikata Kida's Research Papers

  1. (with Robin Tucker-Drob) Groups with infinite FC-center have the Schmidt property,
    preprint, to appear in Ergodic Theory Dynam. Systems. link (open access) arXiv:1901.08735

    Abstract: We show that any countable group with infinite FC-center has the Schmidt property, i.e., admits a free, ergodic, measure-preserving action on a standard probability space such that the full group of the associated orbit equivalence relation contains a non-trivial central sequence. As its consequence, any countable, inner amenable group with property (T) has the Schmidt property.

  2. (with Robin Tucker-Drob) Inner amenable groupoids and central sequences,
    Forum Math. Sigma 8 (2020), e29, 84 pp. link (open access) arXiv:1810.11569

    Abstract: We introduce inner amenability for discrete p.m.p. groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations which either are stable or have a non-trivial central sequence in their full group.

  3. The modular cocycle from commensuration and its Mackey range,
    In: Operator algebras and mathematical physics, 139--152, Adv. Stud. Pure Math., 80, Math. Soc. Japan, Tokyo, 2019. link pdf

    Abstract: The aim of this note is to review construction of the modular cocycle in [17] and provide new examples of groups to which it is applicable, including generalized Baumslag-Solitar groups.

  4. Stable actions and central extensions,
    Math. Ann. 369 (2017), 705--722. link arXiv:1604.04756

    Abstract: A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type II_1 under direct product. We show that for a countable group G and its central subgroup C, if G/C has a stable action, then so does G. Combining a previous result of the author, we obtain a characterization of a central extension having a stable action.

  5. (with Ionut Chifan) OE and W* superrigidity results for actions by surface braid groups,
    Proc. Lond. Math. Soc. (3) 111 (2015), 1431--1470. link arXiv:1502.02391

    Abstract: We show that several important normal subgroups $\Gamma$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $\Gamma \curvearrowright X$ 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.

  6. (with Ionut Chifan and Sujan Pant) Primeness results for von Neumann algebras associated with surface braid groups,
    Int. Math. Res. Not. IMRN 2016, no. 16, 4807--4848. link arXiv:1412.8025

    Abstract: In this paper we introduce a new class of non-amenable groups denoted by $\textbf{NC}_1 \cap \textbf{Quot}(\mathcal{C}_{rss})$ which give rise to prime von Neumann algebras. This means that for every $\Gamma \in \textbf{NC}_1 \cap \textbf{Quot}(\mathcal{C}_{rss})$ its group von Neumann algebra $L(\Gamma)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show $\textbf{NC}_1 \cap \textbf{Quot}(\mathcal{C}_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; all mapping class groups of (punctured) surfaces of genus 0,1,2; most Torelli groups and Johnson kernels of (punctured) surfaces of genus 0,1,2; and, all groups hyperbolic relative to finite families of residually finite, exact, infinite, proper subgroups.

  7. Splitting in orbit equivalence, treeable groups, and the Haagerup property,
    In: Hyperbolic geometry and geometric group theory, 167--214, Adv. Stud. Pure Math., 73, Math. Soc. Japan, Tokyo, 2017. link arXiv:1403.0688

    Abstract: Let $G$ be a discrete countable group and $C$ its central subgroup with $G/C$ treeable. We show that for any treeable action of $G/C$ on a standard probability space $X$, the groupoid $G\ltimes X$ is isomorphic to the direct product of $C$ and $(G/C)\ltimes X$, through cohomology of groupoids. We apply this to show that any group in the minimal class of groups containing treeable groups and closed under taking direct products, commensurable groups and central extensions has the Haagerup property.

  8. Stable actions of central extensions and relative property (T),
    Israel J. Math. 207 (2015), 925--959. link arXiv:1309.3739

    Abstract: Let us say that a discrete countable group is stable if it has an ergodic, free, probability-measure-preserving and stable action. Let G be a discrete countable group with a central subgroup C. We present a sufficient condition and a necessary condition for G to be stable. We show that if the pair (G, C) does not have property (T), then G is stable. We also show that if the pair (G, C) has property (T) and G is stable, then the quotient group G/C is stable.

    Refinement (Mar 2021): Lemma 2.1 for p=1 (that is necessary for proving the main result) can be replaced by a much more elementary argument found in the proof of [27, Lemma 6.2], for comparing L^1 and L^2 norms.

  9. (with Ionut Chifan and Adrian Ioana) W*-superrigidity for arbitrary actions of central quotients of braid groups,
    Math. Ann. 361 (2015), 563--582. link arXiv:1307.5245

    Abstract: For any $n\geqslant 4$ let $\tilde B_n=B_n/Z(B_n)$ be the quotient of the braid group $B_n$ through its center. We prove that any free ergodic probability measure preserving (pmp) action $\tilde B_n\curvearrowright (X,\mu)$ is W$^*$-superrigid in the following sense: if $L^{\infty}(X)\rtimes\tilde B_n\cong L^{\infty}(Y)\rtimes\Lambda$, for an arbitrary free ergodic pmp action $\Lambda\curvearrowright (Y,\nu)$, then the actions $\tilde B_n\curvearrowright X,\Lambda\curvearrowright Y$ are stably (or, virtually) conjugate. Moreover, we prove that the same holds if $\tilde B_n$ is replaced with a finite index subgroup of the direct product $\tilde B_{n_1}\times\cdots\times\tilde B_{n_k}$, for some $n_1,\ldots,n_k\geqslant 4$. The proof uses the dichotomy theorem for normalizers inside crossed products by free groups from \cite{PV11} in combination with the OE superrigidity theorem for actions of mapping class groups from \cite{Ki06}.

  10. Inner amenable groups having no stable action,
    Geom. Dedicata 173 (2014), 185--192. link arXiv:1211.0863

    Abstract: We construct inner amenable and ICC groups having no ergodic, free, probability-measure-preserving and stable action. This solves a problem posed by Jones-Schmidt in 1987.

  11. Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups,
    Groups Geom. Dyn. 9 (2015), 203--235. link arXiv:1205.5123

    Abstract: A measure-preserving action of a discrete countable group on a standard probability space is called stable if the associated equivalence relation is isomorphic to its direct product with the ergodic hyperfinite equivalence relation of type II_1. We show that any Baumslag-Solitar group has such an ergodic, free and stable action. It follows that any Baumslag-Solitar group is measure equivalent to its direct product with any amenable group. The same property is obtained for the inner amenable groups of Vaes.

  12. Invariants of orbit equivalence relations and Baumslag-Solitar groups,
    Tohoku Math. J. (2) 66 (2014), 205--258. link arXiv:1111.3701

    Abstract: To an ergodic, essentially free and measure-preserving action of a non-amenable Baumslag-Solitar group on a standard probability space, a flow is associated. The isomorphism class of the flow is shown to be an invariant of such actions of Baumslag-Solitar groups under weak orbit equivalence. Results on groups which are measure equivalent to Baumslag-Solitar groups are also provided.

  13. (with Saeko Yamagata) Automorphisms of the Torelli complex for the one-holed genus two surface,
    Tokyo J. Math. 37 (2014), 335--372. link arXiv:1009.0568

    Abstract: Let S be a connected, compact and orientable surface of genus two with one boundary component. We study automorphisms of the Torelli complex for S and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for S into itself and show that the Torelli group for S is co-Hopfian.

  14. Examples of amalgamated free products and coupling rigidity,
    Ergodic Theory Dynam. Systems 33 (2013), 499--528. link arXiv:1007.1529

    Abstract: We present amalgamated free products satisfying coupling rigidity with respect to the automorphism group of the associated Bass-Serre tree. As an application, we obtain orbit equivalence rigidity for amalgamated free products of mapping class groups.

  15. (with Saeko Yamagata) The co-Hopfian property of surface braid groups,
    J. Knot Theory Ramifications 22 (2013), 1350055, 46 pp. link arXiv:1006.2599

    Abstract: 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.

  16. (with Saeko Yamagata) Commensurators of surface braid groups,
    J. Math. Soc. Japan 63 (2011), 1391--1435. link arXiv:1004.2946

    Abstract: We prove that when both g and n are integers at least two, 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.

  17. Injections of the complex of separating curves into the Torelli complex,
    preprint. arXiv:0911.3926

    Abstract: We show that for all but finitely many compact orientable surfaces, any superinjective map from the complex of separating curves into the Torelli complex is induced by an element of the extended mapping class group. As an application, we prove that any injective homomorphism from a finite index subgroup of the Johnson kernel into the Torelli group for such a surface is induced by an element of the extended mapping class group.

  18. The co-Hopfian property of the Johnson kernel and the Torelli group,
    Osaka J. Math. 50 (2013), 309--337. link arXiv:0911.3923

    Abstract: For all but finitely many compact orientable surfaces, we show that any superinjective map from the complex of separating curves into itself is induced by an element of the extended mapping class group. We apply this result to proving that any finite index subgroup of the Johnson kernel is co-Hopfian. Analogous properties are shown for the Torelli complex and the Torelli group.

  19. Automorphisms of the Torelli complex and the complex of separating curves,
    J. Math. Soc. Japan 63 (2011), 363--417. link arXiv:0909.4718

    Abstract: We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping class group.

  20. Rigidity of amalgamated free products in measure equivalence,
    J. Topol. 4 (2011), 687--735. link arXiv:0902.2888

    Abstract: A discrete countable group Γ is said to be ME rigid if any discrete countable group that is measure equivalent to Γ is virtually isomorphic to Γ. In this paper, we construct ME rigid groups by amalgamating two groups satisfying rigidity in a sense of measure equivalence. A class of amalgamated free products is introduced, and discrete countable groups which are measure equivalent to a group in that class are investigated.

  21. Measurable rigidity for some amalgamated free products,
    RIMS Kôkyûroku 1627 (2009), 87--98. pdf

    Abstract: This article is a note for my talk at the workshop “Problems in Theory of Operator Algebras” on September 10--12, 2008 at RIMS, Kyoto University.

  22. Introduction to measurable rigidity of mapping class groups,
    In: Handbook of Teichmüller theory, Vol. II, 297--367, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009. link pdf

    Abstract: This is a survey article on my papers ``Measure equivalence rigidity of the mapping class group" and ``Orbit equivalence rigidity for ergodic actions of the mapping class group".

  23. Outer automorphism groups of equivalence relations for mapping class group actions,
    J. Lond. Math. Soc. (2) 78 (2008), 622--638. link

    Abstract: We study outer automorphism groups Out(R) of equivalence relations R arising from ergodic, essentially free and measure-preserving actions on standard probability spaces of mapping class groups, and compute them explicitly for some examples. We show that the subgroups of Out(R) associated with action-automorphisms are always finite index ones. Moreover, a sharp upper bound of the indices among all such actions of mapping class groups is obtained.

  24. Classification of certain generalized Bernoulli actions of mapping class groups,
    preprint (2008). pdf

    Abstract: We present a classification result up to conjugacy of generalized Bernoulli actions of mapping class groups arising geometrically. As a consequence of a rigidity result due to the author, this gives a classication of such actions up to orbit equivalence.

  25. Orbit equivalence rigidity for ergodic actions of the mapping class group,
    Geom. Dedicata 131 (2008), 99--109. link arXiv:math/0607601

    Abstract: We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar results for a finite direct product of mapping class groups as well.

  26. Measure equivalence rigidity of the mapping class group,
    Ann. of Math. (2) 171 (2010), 1851--1901. link arXiv:math/0607600

    Abstract: We show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence: If the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernels. Moreover, we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group. We obtain similar results for finite direct products of mapping class groups.

  27. Classification of the mapping class groups up to measure equivalence,
    Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), 4--7. link

    Abstract: We study the mapping class groups of compact orientable surfaces from the viewpoint of measure equivalence. In this paper, we announce some classification result of the mapping class groups in terms of measure equivalence and the result that there exist various kinds of discrete groups which are not measure equivalent to the mapping class groups. As a byproduct of the proof, it turns out that the mapping class group is exact.

  28. The mapping class group from the viewpoint of measure equivalence theory,
    Mem. Amer. Math. Soc. 196 (2008), no. 916. link arXiv:math/0512230

    Abstract: We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover, we give various examples of discrete groups which are not measure equivalent to the mapping class groups. In the course of the proof, we investigate amenability in a measurable sense for the actions of the mapping class group on the boundary at infinity of the curve complex and on the Thurston boundary. Using this investigation, we prove that the mapping class group of a compact orientable surface is exact.

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