Updated on 2025/04/04

写真a

 
Oguni Shin-Ichi
 
Organization
Graduate School of Science and Engineering (Science) Major of Science and Engineering Mathematics and Computer Science Professor
Title
Professor
Contact information
メールアドレス
External link

Degree

  • 博士（理学） ( 京都大学 )

Professional Memberships

Papers

  • Acylindrical hyperbolicity of Artin groups associated with graphs that are not cones Reviewed

    Motoko Kato, Shin-ichi Oguni

    Groups, Geometry, and Dynamics   18 ( 4 )   1291 - 1316   2024.9

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：European Mathematical Society - EMS - Publishing House GmbH  

    Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them. In this paper, by developing their study and formulating some additional discussion, we demonstrate that acylindrical hyperbolicity holds for more general Artin groups. Indeed, we are able to treat Artin groups of infinite type associated with graphs that are not cones.

    DOI： 10.4171/ggd/783

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  • Coarse compactifications and controlled products Reviewed

    Tomohiro Fukaya, Shin-ichi Oguni, Takamitsu Yamauchi

    Journal of Topology and Analysis   14 ( 04 )   875 - 900   2022.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Ltd  

    We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

    DOI： 10.1142/s1793525321500102

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  • ACYLINDRICAL HYPERBOLICITY OF ARTIN–TITS GROUPS ASSOCIATED WITH TRIANGLE-FREE GRAPHS AND CONES OVER SQUARE-FREE BIPARTITE GRAPHS Reviewed

    MOTOKO KATO, SHIN-ICHI OGUNI

    Glasgow Mathematical Journal   64 ( 1 )   51 - 64   2022.1

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press (CUP)  

    <title>Abstract</title>It is conjectured that the central quotient of any irreducible Artin–Tits group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin–Tits groups that are known to be CAT(0) groups by a result of Brady and McCammond, that is, Artin–Tits groups associated with graphs having no 3-cycles and Artin–Tits groups of almost large type associated with graphs admitting appropriate directions. In particular, the latter family contains Artin–Tits groups of large type associated with cones over square-free bipartite graphs.

    DOI： 10.1017/s0017089520000555

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  • A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture Reviewed

    Tomohiro Fukaya, Shin-ichi Oguni

    Journal of Topology and Analysis   12 ( 03 )   857 - 895   2020.9

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：World Scientific Pub Co Pte Lt  

    We establish a coarse version of the Cartan-Hadamard theorem, which states<br />
    that proper coarsely convex spaces are coarsely homotopy equivalent to the open<br />
    cones of their ideal boundaries. As an application, we show that such spaces<br />
    satisfy the coarse Baum-Connes conjecture. Combined with the result of<br />
    Osajda-Przytycki, it implies that systolic groups and locally finite systolic<br />
    complexes satisfy the coarse Baum-Connes conjecture.

    DOI： 10.1142/s1793525319500675

    arXiv

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  • On relative hyperbolicity for a group and relative quasiconvexity for a subgroup Reviewed

    Yoshifumi Matsuda, Shin-ichi Oguni, Saeko Yamagata

    Tokyo Journal of Mathematics   42   83 - 112   2019.6

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    Language：English   Publishing type：Research paper (scientific journal)  

    We consider two families of subgroups of a group. Each subgroup which belongs<br />
    to one family is contained in some subgroup which belongs to the other family.<br />
    We then discuss relations of relative hyperbolicity for the group with respect<br />
    to the two families, respectively. If the group is supposed to be hyperbolic<br />
    relative to the two families, respectively, then we consider relations of<br />
    relative quasiconvexity for a subgroup of the group with respect to the two<br />
    families, respectively.

    arXiv

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  • On coarse geometric aspects of Hilbert geometry Reviewed

    Ryosuke Mineyama, Shin-ichi Oguni

    Monatshefte fur Mathematik   187 ( 4 )   1 - 16   2018.3

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer-Verlag Wien  

    We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum–Connes conjecture holds for any two-dimensional Hilbert geometry via Yu’s theorem.

    DOI： 10.1007/s00605-018-1171-1

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  • Coronae of relatively hyperbolic groups and coarse cohomologies Reviewed

    Tomohiro Fukaya, Shin-ichi Oguni

    JOURNAL OF TOPOLOGY AND ANALYSIS   8 ( 3 )   431 - 474   2016.9

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

    We construct a corona of a relatively hyperbolic group by blowing-up all parabolic points of its Bowditch boundary. We relate the K-homology of the corona with the K-theory of the Roe algebra, via the coarse assembly map. We also establish a dual theory, that is, we relate the K-theory of the corona with the K-theory of the reduced stable Higson corona via the coarse co-assembly map. For that purpose, we formulate generalized coarse cohomology theories. As an application, we give an explicit computation of the K-theory of the Roe-algebra and that of the reduced stable Higson corona of the fundamental groups of closed 3-dimensional manifolds and of pinched negatively curved complete Riemannian manifolds with finite volume.

    DOI： 10.1142/S1793525316500151

    Web of Science

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  • The coarse Baum-Connes conjecture for Busemann nonpositively curved spaces Reviewed

    Tomohiro Fukaya, Shin-ichi Oguni

    KYOTO JOURNAL OF MATHEMATICS   56 ( 1 )   1 - 12   2016.4

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

    We prove that the coarse assembly maps for proper metric spaces that are nonpositively curved in the sense of Busemann are isomorphisms, where we do not assume that the spaces have bounded coarse geometry. Also it is shown that we can calculate the coarse K-homology and the K-theory of the Roe algebra by using the visual boundaries.

    DOI： 10.1215/21562261-3445129

    Web of Science

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  • The coarse Baum-Connes conjecture and related topics (Japanese) Reviewed

    Shin-ichi Oguni

    Sūgaku   68 ( 2 )   177 - 199   2016.4

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    Language：Japanese  

    CiNii Books

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  • Coronae of product spaces and the coarse Baum-Connes conjecture Reviewed

    Tomohiro Fukaya, Shin-ichi Oguni

    ADVANCES IN MATHEMATICS   279   201 - 233   2015.7

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

    We study the coarse Baum-Connes conjecture for product spaces and product groups. We show that a product of CAT(0) groups, polycyclic groups and relatively hyperbolic groups which satisfy some assumptions on peripheral subgroups, satisfies the coarse Baum-Connes conjecture. For this purpose, we construct and analyze an appropriate compactification and its boundary, "corona", of a product of proper metric spaces. (C) 2015 Elsevier Inc. All rights reserved.

    DOI： 10.1016/j.aim.2015.01.022

    Web of Science

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  • Notes on relatively hyperbolic groups and relatively quasiconvex Subgroups Reviewed

    Yoshifumi Matsuda, Shin-Ichi Oguni, Saeko Yamagata

    Tokyo Journal of Mathematics   38 ( 1 )   99 - 123   2015.6

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：International Academic Printing Co. Ltd.  

    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.

    DOI： 10.3836/tjm/1428412566

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  • The universal relatively hyperbolic structure on a group and relative quasiconvexity for subgroups Reviewed

    Matsuda Yoshifumi, Oguni Shin-ichi, Yamagata Saeko

    RIMS Kokyuroku Bessatsu   48   73 - 93   2014.6

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    Language：English   Publisher：京都大学  

    CiNii Books

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  • On Cannon-Thurston maps for relatively hyperbolic groups Reviewed

    Yoshifumi Matsuda, Shin-ichi Oguni

    JOURNAL OF GROUP THEORY   17 ( 1 )   41 - 47   2014.1

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：WALTER DE GRUYTER GMBH  

    Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon-Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon-Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

    DOI： 10.1515/jgt-2013-0024

    Web of Science

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  • Dilatational equivalence classes, Novikov-Shubin type capacities of groups, and random walks Reviewed

    Shin-ichi Oguni

    Noncommutative Geometry and Physics, 3   433 - 469   2013.3

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    Language：English  

    We have two purposes in this paper. One is to define and to study dilatational equivalence classes and Novikov-Shubin type capacities of arbitrary discrete groups from an algebraic viewpoint. The other is to refine some known results about asymptotic behaviors of return probabilities of simple symmetric random walks on finitely generated groups. Also this paper contains an appendix where we give a remark on a paper by Gromov and Shubin.

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  • C*-SIMPLICITY FOR GROUPS WITH NON-ELEMENTARY CONVERGENCE GROUP ACTIONS Reviewed

    Yoshifumi Matsuda, Shin-ichi Oguni, Saeko Yamagata

    HOUSTON JOURNAL OF MATHEMATICS   39 ( 4 )   1291 - 1299   2013

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV HOUSTON  

    We prove that a countable group with an effective minimal non-elementary convergence group action is a Powers group. More strongly we prove that it is a strongly Powers group and thus its non-trivial subnormal subgroups are C*-simple.

    Web of Science

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  • THE COARSE BAUM-CONNES CONJECTURE FOR RELATIVELY HYPERBOLIC GROUPS Reviewed

    Tomohiro Fukaya, Shin-Ichi Oguni

    JOURNAL OF TOPOLOGY AND ANALYSIS   4 ( 1 )   99 - 113   2012.3

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

    We study a group which is hyperbolic relative to a finite family of infinite subgroups. We show that the group satisfies the coarse Baum-Connes conjecture if each subgroup belonging to the family satisfies the coarse Baum-Connes conjecture and admits a finite universal space for proper actions. If the group is torsion-free, then it satisfies the analytic Novikov conjecture.

    DOI： 10.1142/S1793525312500021

    Web of Science

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  • Secondary Novikov-Shubin invariants of groups and quasi-isometry Reviewed

    Shin-ichi Oguni

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   59 ( 1 )   223 - 237   2007.1

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

    We define new L-2-invariants which we call secondary Novikov-Shubin invariants. We calculate the first secondary Novikov-Shubin invariants of finitely generated groups by using random walk on Cayley graphs and see in particular that these are invariant under quasi-isometry.

    DOI： 10.2969/jmsj/1180135508

    Web of Science

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  • The group homology and an algebraic version of the zero-in-the-spectrum conjecture Reviewed

    Shin-ichi Oguni

    JOURNAL OF MATHEMATICS OF KYOTO UNIVERSITY   47 ( 2 )   359 - 369   2007

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KINOKUNIYA CO LTD  

    We introduce an algorithm which transforms a finitely presented group G into another one G(Psi). By using this, we can get many finitely presented groups whose group homology with coefficients in the group von Neumann algebra vanish, that is, many counterexamples to an algebraic version of the zero-in-the-spectrum conjecture. Moreover we prove that the Baum-Connes conjecture does not imply the algebraic version of the zero-in-the-spectrum conjecture for finitely presented groups. Also we will show that for any p &gt;= 3 the p-th group homology of G(Psi) coming from free groups has infinite rank.

    Web of Science

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MISC

  • Acylindrical hyperbolicity and the centers of Artin groups that are not free of infinity

    Motoko Kato, Shin-ichi Oguni

    preprint   2024.6

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  • Blowing up and down compacta with geometrically finite convergence actions of a group

    Yoshifumi Matsuda, Shin-ichi Oguni, Saeko Yamagata

    2012.1

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    Publishing type：Internal/External technical report, pre-print, etc.  

    We consider two compacta with minimal non-elementary convergence actions of a<br />
    countable group. When there exists an equivariant continuous map from one to<br />
    the other, we call the first a blow-up of the second and the second a blow-down<br />
    of the first. When both actions are geometrically finite, it is shown that one<br />
    is a blow-up of the other if and only if each parabolic subgroup with respect<br />
    to the first is parabolic with respect to the second. As an application, for<br />
    each compactum with a geometrically finite convergence action, we construct its<br />
    blow-downs with convergence actions which are not geometrically finite.

    arXiv

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  • Hyperbolically embedded virtually free subgroups of relatively hyperbolic groups

    Yoshifumi Matsuda, Shin-ichi Oguni, Saeko Yamagata

    2011.9

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    Publishing type：Internal/External technical report, pre-print, etc.  

    We show that if a group is not virtually cyclic and is hyperbolic relative to<br />
    a family of proper subgroups, then it has a hyperbolically embedded subgroup<br />
    which contains a finitely generated non-abelian free group as a finite index<br />
    subgroup.

    arXiv

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  • L2-invariants of groups under coarse equivalence and of groupoids under Morita equivalence

    Shin-ichi Oguni

    preprint   2010.8

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    We prove that triviality of some L2-invariants of discrete groups is preserved by coarse equivalence, where L2-invariants are L2-homologies, L2-Betti numbers (with `a mild condition&#039;) and Novikov-Shubin invariants. We give definitions of some L2-invariants of cocompact etale groupoids and prove that their triviality is preserved by Morita equivalence. Also we exhibit basic properties for modules over von Neumann algebras which are not necessarily finite. <br />
    This paper contains an appendix by Yamashita, where a characterization of finite von Neumann algebras is given.

    File： oguni110501.pdf

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Presentations

  • アルティン群の非シリンダー的双曲性 Invited

    尾國新一

    第71回 幾何学シンポジウム  2024.9 

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    Event date： 2024.9

    Language：Japanese   Presentation type：Oral presentation (keynote)  

    Venue：関西大学  

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  • 完全グラフでない定義グラフに付随したアルティン群の非シリンダー的双曲性

    尾國新一

    2024 日本数学会 秋季総合分科会  2024.9 

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    Event date： 2024.9

    Presentation type：Oral presentation (general)  

    Venue：大阪大学  

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  • 無限型のアルティン群の非シリンダー的双曲性について

    尾國新一

    RIMS共同研究 : 変換群論の新潮流  2022.5 

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    Event date： 2022.5

    Language：Japanese   Presentation type：Oral presentation (general)  

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  • 粗Baum-Connes予想と非正曲率性

    尾國新一

    代数的位相幾何学の軌跡と展望～山崎正之先生退職祝賀研究集会～  2022.3 

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    Event date： 2022.3

    Language：Japanese   Presentation type：Oral presentation (general)  

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  • 無限型のアルティン群の非シリンダー的双曲性について

    尾國新一

    早稲田双曲幾何幾何学的群論セミナー  2021.12 

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    Event date： 2021.12

    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：オンライン  

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  • 無限型のアルティン群の非シリンダー的双曲性について

    尾國新一

    東工大トポロジーセミナー  2021.11 

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    Event date： 2021.11

    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：オンライン  

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  • Artin-Tits群の非シリンダー的双曲性について

    尾國新一

    一般位相幾何学とその関連分野の進展  2020.10 

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    Event date： 2020.10

    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：オンライン  

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  • Artin群の非シリンダー的双曲性について

    尾國新一

    第15回代数・解析・幾何学セミナー  2020.2 

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    Event date： 2020.2

    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：鹿児島大学   Country：Japan  

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  • Osajdaによるエクスパンダーを含む群とその周辺

    尾國新一

    春の代数的位相幾何学セミナー~幾何的アプローチ  2016.3 

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    Venue：岡山理科大学(岡山県岡山市)  

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  • Hilbert幾何の粗幾何的性質

    尾國新一

    白浜研究集会  2016.3 

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    Language：Japanese   Presentation type：Oral presentation (general)  

    Venue：南紀白浜温泉旅館むさし(和歌山県西牟婁郡)  

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  • 粗ホモトピーについて

    尾國新一

    第3回幾何学的群論若手勉強会  2017.3 

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    Venue：ルネッサ赤沢(静岡県伊東市)  

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  • On coarse homotopy

    Shin-ichi Oguni

    Research Trends on Set-theoretic and Geometric Topology and their cooperation with various branches  2017.6 

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    Venue：RIMS(Kyoto)  

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  • 粗Baum-Connes予想と粗代数的トポロジー Invited

    尾國新一

    日本数学会2014年度秋季総合分科会  2014.9 

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    Presentation type：Oral presentation (invited, special)  

    Venue：広島大学(広島県東広島市)  

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  • 非有界距離空間の大尺度幾何と無限遠

    尾國 新一

    松山TGSAセミナー  2015.6 

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    Venue：愛媛大学(松山)  

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  • 粗Baum-Connes予想と粗幾何

    尾國新一

    広島幾何学研究集会2015  2015.10 

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    Venue：広島大学(広島県東広島市)  

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  • Coronae of product spaces and the coarse Baum-Connes conjecture

    Shin-ichi Oguni

    Conference on Non-commutative Geometry and K-Theory  2015.12 

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    Venue：Chongqing University, China  

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  • An introduction to the coarse Baum-Connes conjecture

    尾國新一

    非可換幾何 湯谷研究集会  2013.11 

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    Venue：湯谷温泉-湯の風HAZU-(愛知県新城市)  

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  • Coronae and coarse homologies

    Shin-ichi Oguni

    Metric geometry and analysis  2013.12 

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    Venue：Kyoto university  

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  • Coarse Baum-Connes conjecture and coarse algebraic topology

    Shin-ichi Oguni

    Rigidity School, Tokyo 2013/2014  2014.1 

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    Venue：Tokyo university  

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  • 粗幾何版アダマール・カルタンの定理について

    尾國 新一

    日本数学会2018年度年会  2018.3 

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    Venue：東京大学  

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  • A coarse Cartan-Hadamard theorem with application to the coarse Baum-Connes conjecture Invited

    Shin-ichi Oguni

    Asia-Pacific GNCG Seminar  2021.4 

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    Language：English   Presentation type：Oral presentation (general)  

    Venue：online  

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  • On a coarse Cartan-Hadamard theorem

    Shin-ichi Oguni

    Rigidity School ― The Final Meeting  2018.9 

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    Venue：Nagoya university  

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  • Coarsely convex spaces and a coarse Cartan-Hadamard theorem

    尾國 新一

    トポロジー火曜セミナー（東大数理）  2018.11 

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    Venue：東京大学  

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  • On a coarse Cartan-Hadamard theorem Invited International conference

    Shin-ichi Oguni

    AMS Sectional meeting  2019.3 

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    Language：English  

    Venue：University of Hawaii at Manoa, Honolulu, HI  

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  • 粗幾何版アダマール・カルタンの定理とその応用

    尾國 新一

    松山TGSAセミナー  2017.10 

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    Venue：愛媛大学(松山)  

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  • 粗凸距離空間入門

    尾國 新一

    ワークショップ：幾何学的群論の新展開  2018.2 

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    Venue：ルネッサ赤沢(静岡県伊東市)  

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  • On the coarse Baum-Connes conjecture

    Shin-ichi Oguni

    GEOQUANT 2013  2013.8 

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    Venue：Erwin-Schrodinger International Institute for Mathematical Physics (ESI), Vienna, Austria  

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Research Projects

  • 粗Baum-Connes予想に関わる粗幾何学の新展開

    2020.4 - 2025.3

    日本学術振興会  科学研究費助成事業 基盤研究(C)  基盤研究(C)

    尾國 新一

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    Grant amount：\4550000 （ Direct Cost: \3500000 、 Indirect Cost：\1050000 ）

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  • 粗Baum-Connes予想に関わる粗幾何学

    2016.4 - 2020.3

    JSPS  Grant-in-Aid for Young Scientists (B) 

    OGUNI Shin-ichi

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    Authorship：Principal investigator  Grant type：Competitive

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  • Moduli theory of non linear elliptic operators over non compact manifolds

    2013.4 - 2017.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)

    Kato Tsuyoshi, Kida Yoshikata, Oguni Shin-ichi, Fukaya Tomohiro, Tsukamoto Masaki, Matsuo Shinichiroh

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    Grant amount：\15600000 （ Direct Cost: \12000000 、 Indirect Cost：\3600000 ）

    I have constructed a monopole map over the universal covering space of a compact oriented smooth four manifold. We apply the infinite dimensional Bott periodicity by Higson-Kasparov-Trout. In particular its degree was given when the linearized map is isomorphic, as an element in the equivariant E theory. It produces a homomorphism between K group of C* algebras related to the group ring. It corresponds to a covering version of the Bauer-Furuta degree.
    As an application, we proposed an aspherical 10/8 inequality for spin classifying 4 manifolds. We have also verified that it certainly holds for large class of 4 manifolds which includes complex minimal surfaces of general type.

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  • 離散群の相対的双曲構造と収束群作用

    2012.4 - 2016.3

    JSPS  Grant-in-Aid for Young Scientists (B) 

    OGUNI Shin-ichi

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    Authorship：Principal investigator  Grant type：Competitive

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  • 離散群の相対双曲構造

    2011.4 - 2012.3

    愛媛大学  理学部長裁量研究助成費 

    OGUNI Shin-ichi

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    Authorship：Principal investigator  Grant type：Competitive

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  • 普遍的な幾何学的有限収束作用を持つ離散群 の探求とそのような離散群の性質の研究

    2010.4 - 2011.3

    愛媛大学  理学部長裁量研究助成費 

    OGUNI Shin-ichi

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    Authorship：Principal investigator  Grant type：Competitive

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  • 非可換幾何的視点からの離散群とエタール亜群の研究

    2008

    日本学術振興会  科学研究費助成事業  特別研究員奨励費

    尾國 新一

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    Grant amount：\800000 （ Direct Cost: \800000 ）

    私の研究課題である「非可換幾何的視点からの離散群とエタール亜群の研究」に基づき、今年度、得られた結果について書く.なお、これに関する論文は執筆中であるが口頭では発表した。可算離散群には,coarse equivalenceを除いて一意的に,群作用不変な固有距離が入ることがよく知られており,二つの可算離散群が互いにcoarsely equivalentであるというのは,それぞれに,群作用不変な固有距離を入れた距離空間として,互いにcoarsely equivalentであると定義される.可算離散群に対して,coarsely equivalent不変な量や概念およびcoarsely equivalentのもとでの分類などを研究することが幾何学的群論と呼ばれるものである.ところで,二つの可算離散群が互いにcoarsely equivalentであることは,ある二つの擬群が森田同値であることだと理解できることが考察できる(M.Gromovの定理の変種),このcoarse equivalenceの言い換えは,幾何学的群論の研究においては,あまり利用されていない孤立した結果であったように思う.しかし,以下の定理を示す際には,非常に強力な手段として用いた.
    定理二つの可算離散群が互いにcoarsely equivalentであるとする.
    このとき,一方のL2不変量が自明ならば,もう一方のそれも自明である.
    ここで、L2不変量とはNovikov-Shubin不変量やL2ベッチ数を意味する.これらの結果は,群により強い制限をおいた元では,Pansuなどにより知られており,それを一般的な場合に拡張したものになっている.一方,証明方法は彼のものとはまったく異なり,私の使った定理の証明法は,擬群論の発展を同時に促すものであることも優位な点である.

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  • ラプラシアンのスペクトラル密度関数を用いて、多様体とその基本群を研究する。

    2005 - 2007

    日本学術振興会  科学研究費助成事業  特別研究員奨励費

    尾國 新一

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    Grant amount：\2700000 （ Direct Cost: \2700000 ）

    群フォンノイマン環NG係数の群ホモロジーのようなNG加群の大きさ(小ささ)を測るものとして1^2不変量がある.特に1^2ベッチ数はNG係数の群ホモロジーの射影部分加群の大きさを測っており,一方で,可測と呼ばれている1^2ベッチ数が消えている部分加群の小ささをNS不変量が測っている.ところで,・1^2ベッチ数ば射影加群に対して忠実な量であるのに対して,NS不変量や私が以前に定義した二義的NS不変量(これらをまとめて,NS型不変量と呼ぶことにする)は有限表示可測加群に対して,忠実ではない.つまり,0でない有限表示可測加群に対して,NS型不変量が無限になってしまうことがある.一方で,有限表示可測加群に対してNS型不変量を定義する際に経由するスペクトラル密度関数という実数値ではない不変量は,忠実である.そこで,わたしは一般の有限表示とは限らない可測加群に対してもスペクトラル密度関数を定義し,忠実となるものを定義した.そのためのアイデアはスペクトラル密度関数の代わりにスペクトラル密度関数の族を考えるということである.この概念を用いて,いくつかの知られている定理がきれいに一般化できたり,証明の見通しを良くできることを注意しておく.例えば,有限生成とは限らない離散群上のランダムウォークについていくつか定理が得られた.有限生成群の場合に限っても、知られていたいくつかの定理にコンセプチュアルな別証明を与えることができた.この研究は,研究の目的をより高い視点に立って,行ったものである.

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