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Tetsu Toyoda

Bio: Tetsu Toyoda is an academic researcher from Nagoya University. The author has contributed to research in topics: Metric space & Fixed point. The author has an hindex of 4, co-authored 11 publications receiving 36 citations.

Papers
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Journal ArticleDOI
01 Mar 2012
TL;DR: In this paper, it was shown that a random group has fixed points when it isometrically acts on a CAT(0) cube complex, assuming that the action is simplicial.
Abstract: We prove that a random group has fixed points when it isometrically acts on a CAT(0) cube complex. We do not assume that the action is simplicial.

9 citations

Journal ArticleDOI
Tetsu Toyoda1
TL;DR: In this paper, the Izeki-Nayatani invariants of spaces in such a family are uniformly bounded from above by a constant strictly less than 1, and the invariants are shown to be invariant to the tangent cones of the Gromov-Hausdorff precompact family of spaces.
Abstract: In this paper, we will consider a family $\mathscr{Y}$ of complete CAT(0) spaces such that the tangent cone TCp Y at each point p $\in$ Y of each Y $\in$ $\mathscr{Y}$ is isometric to a (finite or infinite) product of the Euclidean cones Cone(Xα) over elements Xα of some Gromov-Hausdorff precompact family {Xα} of CAT(1) spaces. Each element of such $\mathscr{Y}$ is a space presented by Gromov [4] as an example of a "CAT(0) space with "bounded" singularities". We will show that the Izeki-Nayatani invariants of spaces in such a family are uniformly bounded from above by a constant strictly less than 1.

8 citations

Posted Content
Tetsu Toyoda1
TL;DR: In this paper, a geometric condition for a family of CAT(0) spaces is presented, which ensures that the Izeki-Nayatani invariants of spaces in the family are uniformly bounded from above by a constant strictly less than 1.
Abstract: In this paper, we present a geometric condition for a family of CAT(0) spaces, which ensures that the Izeki-Nayatani invariants of spaces in the family are uniformly bounded from above by a constant strictly less than 1. Each element of such a family with this condition is a space presented by M. Gromov as an example of a "CAT(0) space with "bounded" singularities". Combining our result with a result of Izeki, Kondo and Nayatani, we see that random groups of Gromov's graph model have a fixed-point property for such a family.

5 citations

Posted Content
TL;DR: In this paper, it was shown that for general metric spaces, Gromov's condition implies the condition for all non-geodesic metric spaces that satisfy the $n\geq 5$ condition.
Abstract: Let $\kappa$ be a real number. Gromov (2001) introduced the $\mathrm{Cycl}_n (\kappa )$ conditions for all integers $n\geq 4$, each of which is a necessary condition for a metric space to admit an isometric embedding into a $\mathrm{CAT}(\kappa )$ space. We prove an analogue of Reshetnyak's majorization theorem for (possibly non-geodesic) metric spaces that satisfy the $\mathrm{Cycl}_4 (\kappa )$ condition. It follows from our result that for general metric spaces, the $\mathrm{Cycl}_4 (\kappa )$ condition implies the $\mathrm{Cycl}_n (\kappa )$ conditions for all integers $n\geq 5$.

5 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the Izeki-Nayatani invariance of a graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.
Abstract: We prove that if a geodesically complete CAT(0) space X admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of X is less than 1. Let G be a finite connected graph, μ1(G) be the linear spectral gap of G, and λ1(G,X) be the nonlinear spectral gap of G with respect to such a CAT(0) space X. Then, the result implies that the ratio λ1(G,X)/μ1(G) is bounded from below by a positive constant which is independent of the graph G. It follows that any isometric action of a random group of the graph model on such X has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.

4 citations


Cited by
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Journal ArticleDOI
Takefumi Kondo1
TL;DR: In this paper, the authors constructed a CAT(0) space Y with Izeki-Nayatani invariant δ(Y) = 1, and proved that there exists a space Y which does not have Markov type p for every p > 1.
Abstract: We construct a CAT(0) space Y with Izeki–Nayatani invariant δ(Y) = 1. By a similar construction, we also prove that there exists a CAT(0) space which does not have Markov type p for every p > 1.

26 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature, as shown by Andoni et al. (2018).
Abstract: We prove that not every metric space embeds coarsely into an Alexandrov space of nonpositive curvature. This answers a question of Gromov (Geometric group theory, Cambridge University Press, Cambridge, 1993) and is in contrast to the fact that any metric space embeds coarsely into an Alexandrov space of nonnegative curvature, as shown by Andoni et al. (Ann Sci Ec Norm Super (4) 51(3):657–700, 2018). We establish this statement by proving that a metric space which is q-barycentric for some $$q\in [1,\infty )$$ has metric cotype q with sharp scaling parameter. Our proof utilizes nonlinear (metric space-valued) martingale inequalities and yields sharp bounds even for some classical Banach spaces. This allows us to evaluate the bi-Lipschitz distortion of the $$\ell _\infty $$ grid $$[m]_\infty ^n=(\{1,\ldots ,m\}^n,\Vert \cdot \Vert _\infty )$$ into $$\ell _q$$ for all $$q\in (2,\infty )$$ , from which we deduce the following discrete converse to the fact that $$\ell _\infty ^n$$ embeds with distortion O(1) into $$\ell _q$$ for $$q=O(\log n)$$ . A rigidity theorem of Ribe (Ark Mat 14(2):237–244, 1976) implies that for every $$n\in {\mathbb {N}}$$ there exists $$m\in {\mathbb {N}}$$ such that if $$[m]_\infty ^n$$ embeds into $$\ell _q$$ with distortion O(1), then q is necessarily at least a universal constant multiple of $$\log n$$ . Ribe’s theorem does not give an explicit upper bound on this m, but by the work of Bourgain (Geometrical aspects of functional analysis (1985/86), Springer, Berlin, 1987) it suffices to take $$m=n$$ , and this was the previously best-known estimate for m. We show that the above discretization statement actually holds when m is a universal constant.

16 citations

Journal ArticleDOI
TL;DR: In this article, a polynomial estimate of all integer Sobolev norms in terms of the weighted norms of the nonlinear Fourier transformed is provided. But the analysis of the regularity properties of the Fourier coefficients is limited.
Abstract: The defocusing NLS-equation $\mathrm{i} u_t = -u_{xx} + 2|u|^2u$ on the circle admits a global nonlinear Fourier transform, also known as Birkhoff map, linearising the NLS-flow. The regularity properties of $u$ are known to be closely related to the decay properties of the corresponding nonlinear Fourier coefficients. In this paper we quantify this relationship by providing two sided polynomial estimates of all integer Sobolev norms $\|u\|_m$, $m\ge 0$, in terms of the weighted norms of the nonlinear Fourier transformed.

12 citations

Journal ArticleDOI
Tetsu Toyoda1
TL;DR: In this article, a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities.
Abstract: Abstract Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a CAT(0) space by modifying and generalizing Gromov’s cycle conditions. Furthermore, we prove that if a metric space satisfies all those necessary conditions, then it admits an isometric embedding into a CAT(0) space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a CAT(0) space.

10 citations

Journal ArticleDOI
01 Mar 2012
TL;DR: In this paper, it was shown that a random group has fixed points when it isometrically acts on a CAT(0) cube complex, assuming that the action is simplicial.
Abstract: We prove that a random group has fixed points when it isometrically acts on a CAT(0) cube complex. We do not assume that the action is simplicial.

9 citations