Papers - Izeki, Hiroyasu

Torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point

Hiroyasu Izeki, Anders Karlsson

Commentarii Mathematici Helvetici (EMS Press)   2026.04

Research paper (scientific journal), Joint Work, Accepted

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We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin, Osajda, and Przytycki. It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks.

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A CAT(0) alternative for amenable groups and a Kazhdan-type rigidity principle

Hiroyasu Izeki, Ran Ji, Anders Karlsson, Yunhui Wu

 2026.03

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We prove that finitely generated amenable groups acting on CAT(0) spaces satisfy the following alternative: either every action on a geodesically complete CAT(0) space with bounded geometry (or finite dimension) has a global fixed point, or the group admits a fixed-point-free action on $\mathbb{R}^n$. As a consequence, finitely generated amenable torsion groups and finitely generated virtually simple amenable groups cannot act nontrivially on geodesically complete CAT(0) spaces with bounded geometry or on finite-dimensional complete CAT(0) spaces.
The proof relies on a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property: if such a group acts on a geodesically complete CAT(0) space of bounded geometry with almost fixed points, then it has a genuine fixed point. This yields several further corollaries, including a rigidity dichotomy for drift and that any finitely generated torsion group acting on a geodesically complete visibility CAT(0) space with bounded geometry must have a global fixed point. These results make substantial progress on the longstanding problem of understanding actions of torsion groups on CAT(0) spaces.

Isometric group actions with vanishing rate of escape on CAT(0) spaces

Hiroyasu Izeki

Geom. Funct. Anal.  33 ( 1 ) 170 - 244 2023.01

Research paper (scientific journal), Single Work, Accepted

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A fixed point property of random groups

IZEKI Hiroyasu

京都大学数理解析研究所講究録  2018.04

Research paper (conference, symposium, etc.), Single Work

Poisson boundary and rigidity of discrete groups

IZEKI Hiroyasu

Geometry and Analysis, Fukuoka 1   117 - 122 2015.12

Research paper (conference, symposium, etc.), Single Work

Fixed-point property of random quotients by plain words

Hiroyasu Izeki

Groups, Geometry, and Dynamics (EUROPEAN MATHEMATICAL SOC)  8 ( 4 ) 1101 - 1140 2014.12

Research paper (scientific journal), Single Work, Lead author, Last author, Corresponding author, Accepted,  ISSN  1661-7207

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We show a fixed-point property of certain random groups for a wide class of CAT(0) spaces. The model of random groups under consideration is given as the set of presentations (S, R), where S is a generating set and the set of relations R is a subset of the set of all plain words of the same length with suitably fixed density. Our main theorem says that, with high probability, groups obtained by such presentations have the fixed-point property for all CAT(0) spaces having bounded singularities.

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N-step energy of maps and the fixed-point property of random groups

Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani

GROUPS GEOMETRY AND DYNAMICS (EUROPEAN MATHEMATICAL SOC)  6 ( 4 ) 701 - 736 2012

Research paper (scientific journal), Joint Work, Accepted,  ISSN  1661-7207

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We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT (0) spaces. We use Gromov's criterion for the fixed-point property in terms of the growth of n-step energy of equivariant maps from a finitely generated group into a CAT.(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL (m, Q(r)), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with m bounded from above.

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A fixed-point property of finitely generated groups and an energy of equivariant maps

IZEKI Hiroyasu

"Probabilistic Approach to Geometry", ASPM (MATH SOC JAPAN)  57   171-188 - 188 2010

Research paper (conference, symposium, etc.), Single Work, Accepted

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The purpose of this article is to present sufficient conditions for an isometric action of a finitely generated group to have a fixed point in terms of an energy of equivariant maps (Proposition 2.1 and Theorem 3.1), which turn out to be useful in proving that groups with fixed-point property form a large family in the set of finitely presented groups. Most of the results are included in [7] and [8].

An approach to superrigidity and fixed-point theorems via harmonic maps

IZEKI Hiroyasu, Shin Nayatani

Amer. Math. Soc. Transl. (2) 230   135-160 2010

Research paper (scientific journal), Joint Work, Accepted

Fixed-point property of random groups

IZEKI Hiroyasu, KONDO Takefumi, NAYATANI Shin

Annals of Global Analysis and Geometry (SPRINGER)  35 ( 4 ) 363-379 - 379 2009.06

Research paper (scientific journal), Joint Work, Accepted,  ISSN  0232-704X

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In this paper, using the generalized version of the theory of combinatorial harmonic maps, we give a criterion for a finitely generated group Gamma to have the fixed-point property for a certain class of Hadamard spaces, and prove a fixed-point theorem for random-group actions on the same class of Hadamard spaces. We also study the existence of an equivariant energy-minimizing map from a Gamma-space to the limit space of a sequence of Hadamard spaces with the isometric actions of a finitely generated group Gamma. As an application, we present the existence of a constant which may be regarded as a Kazhdan constant for isometric discrete-group actions on a family of Hadamard spaces.

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A fixed-point theorem for discrete-group actions on Hadamard spaces(General and Geometric Topology and Geometric Group Theory)

Izeki Hiroyasu, Kondo Takefumi, Nayatani Shin

RIMS Kokyuroku (Kyoto University)  1492   56 - 64 2006.05

Research paper (conference, symposium, etc.), Joint Work,  ISSN  1880-2818

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調和写像による超剛性定理および固定点定理へのアプローチ

井関 裕靖, 納谷 信

数学 58   239 - 262 2006

Research paper (scientific journal), Joint Work, Accepted

Combinatorial harmonic maps and discrete-group actions on Hadamard spaces

IZEKI Hiroyasu, NAYATANI Shin

Geometry Dedicata (SPRINGER)  114 ( 1 ) 147-188 - 188 2005.08

Research paper (scientific journal), Joint Work, Accepted,  ISSN  0046-5755

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In this paper we use the combinatorial harmonic map theory to study the isometric actions of discrete groups on Hadamard spaces. Given a finitely generated group acting by automorphisms, properly discontinuously and cofinitely on a simplicial complex and its isometric action on a Hadamard spaces, we formulate criterions for the action to have a global fixed point.

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組合せ調和写像と超剛性 ---singular taget の場合

井関 裕靖, 納谷 信

数理解析研究所講究録 1329   1 - 7 2003

Research paper (conference, symposium, etc.), Joint Work

A fixed-point theorem for discrete-group actions on Hadamard spaces

Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani

数理解析研究所講究録 1492   56 - 64 2003

Research paper (conference, symposium, etc.), Joint Work

Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature

IZEKI Hiroyasu

Proceedings of American Mathematical Society (AMER MATHEMATICAL SOC)  130 ( 12 ) 3731-3740 - 3740 2002.12

Research paper (scientific journal), Single Work, Accepted,  ISSN  0002-9939

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We give a suffcient condition for a higher dimensional Kleinian group Gamma subset of Isom(Hn+1) to be convex cocompact in terms of the critical exponent of. As a consequence, we see that the fundamental group of a compact conformally at manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally at manifolds with positive scalar curvature.

組合せ調和写像と超剛性

井関 裕靖, 納谷 信

数理解析研究所講究録 1270   182 - 194 2002

Research paper (conference, symposium, etc.), Joint Work

高次元のクライン群の極限集合のハウスドルフ次元---収束指数と凸コンパクト性---

井関 裕靖

数理解析研究所講究録 1223   61 - 68 2001

Research paper (conference, symposium, etc.), Single Work

Quasiconformal stability of Kleinian groups and an embedding of a space of flat conformal structures

IZEKI Hiroyasu

Conformal Geometry and Dynamics 4 ( 6 ) 108-119 - 119 2000.12

Research paper (scientific journal), Single Work, Accepted,  ISSN  1088-4173

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We show the quasiconformal stability for torsion-free convex cocompact Kleinian groups acting on higher dimensional hyperbolic spaces. As an application, we prove an embedding theorem of a space of at conformal structures on a certain class of compact manifolds. © 1999 American Mathematical Society.

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The Teichmüller distance on the space of flat conformal structures

Hiroyasu Izeki

Conformal Geometry and Dynamics 2 ( 1 ) 1 - 24 1998.02

Research paper (scientific journal), Single Work, Lead author, Last author, Corresponding author, Accepted,  ISSN  1088-4173

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We define the Teichmüller pseudodistance on spaces of flat conformal structures by the same manner as classical Teichmüller distance on the Teichmüller space of Riemann surfaces. We will prove that for compact manifolds this pseudodistance becomes a complete distance. We will also prove similar results for noncompact manifolds under certain assumptions. © 1998 American Mathematical Society.

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