arXiv subjects
Publications and source records attributed to Tetsu Toyoda.
We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{CAT}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{CAT}(0)$ spaces. In particular, the validity of these inequalities does not follow from the $\mathrm{CAT}(0)$ $4$-point condition.
Andoni, Naor and Neiman (2018) established a family of quadratic metric inequalities that hold true in every CAT(0) space. As stated in their paper, this family seems to include all previously used quadratic metric inequalities that hold true in every CAT(0) space. We prove that there exists a metric space that satisfies all inequalities in this family but does not admit an isometric embedding into any CAT(0) space. More precisely, we prove that the 6-point metric space constructed by Nina Lebedeva, which does not admit an isometric embedding into any CAT(0) space, satisfies all inequalities in this family.
For any real number $κ$ and any integer $n\geq 4$, the $\mathrm{Cycl}_n (κ)$ condition introduced by Gromov (2001) is a necessary condition for a metric space to admit an isometric embedding into a $\mathrm{CAT}(κ)$ space. It is known that for geodesic metric spaces, the $\mathrm{Cycl}_4 (κ)$ condition is equivalent to being $\mathrm{CAT}(κ)$. In this paper, we prove an analogue of Reshetnyak's majorization theorem for (possibly non-geodesic) metric spaces that satisfy the $\mathrm{Cycl}_4 (κ)$ condition. It follows from our result that for general metric spaces, the $\mathrm{Cycl}_4 (κ)$ condition implies the $\mathrm{Cycl}_n (κ)$ conditions for all integers $n\geq 5$, although Gromov stated that this implication is apparently not true.
Gromov (2001) and Sturm (2003) proved that any four points in a $\mathrm{CAT}(0)$ space satisfy a certain family of inequalities. We call those inequalities the $\boxtimes$-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 $\mathrm{CAT}(0)$ space if and only if any four points in $X$ satisfy the $\boxtimes$-inequalities. To prove this, we introduce a new family of necessary conditions for a metric space to admit an isometric embedding into a $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ space. This work presents a new approach to characterizing those metric spaces that admit an isometric embedding into a $\mathrm{CAT}(0)$ space.
We prove that if a geodesically complete $\mathrm{CAT}(0)$ 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, $μ_1 (G)$ be the linear spectral gap of $G$, and $λ_1 (G,X)$ be the nonlinear spectral gap of $G$ with respect to such a $\mathrm{CAT}(0)$ space $X$. Then, the result implies that the ratio $λ_1 (G,X) / μ_1 (G)$ 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.
By generalizing the path method, we show that nonlinear spectral gaps of a finite connected graph are uniformly bounded from below by a positive constant which is independent of the target metric space. We apply our result to an $r$-ball $T_{d,r}$ in the $d$-regular tree, and observe that the asymptotic behavior of nonlinear spectral gaps of $T_{d,r}$ as $r\to\infty$ does not depend on the target metric space, which is in contrast to the case of a sequence of expanders. We also apply our result to the $n$-dimensional Hamming cube $H_n$ and obtain an estimate of its nonlinear spectral gap with respect to an arbitrary metric space, which is asymptotically sharp as $n\to\infty$.
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.
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.