CAT(0) spaces and expanders

Kondo, Takefumi

doi:10.1007/s00209-011-0866-y

Cited by 45 publications

(35 citation statements)

References 22 publications

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“…Kondo [8] constructed the first examples of such spaces. On the other hand, the present author [13] proved the following criterion for a complete CAT(0) space X to be δ(X) < 1 (see Theorem 5.4 in [13]).…”

Section: Izeki-nayatani Invariantmentioning

confidence: 99%

Fixed point property for a CAT(0) space which admits a proper cocompact group action

Toyoda¹

2016

Kodai Math. J.

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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, 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 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.

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Section: Izeki-nayatani Invariantmentioning

confidence: 99%

Fixed point property for a CAT(0) space which admits a proper cocompact group action

Toyoda¹

2016

Kodai Math. J.

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“…Following Gromov [Gro01,Gro03] and Kondo [Kon12], we shall now describe a simple way to obtain CAT (1) spaces from graphs. Suppose that F is a family of connected graphs that satisfies…”

Section: Preliminariesmentioning

confidence: 99%

“…We were motivated to revisit the question of [NS11] that Theorem 1.1 answers by the recent work of Kondo [Kon12]. Kondo's goal in [Kon12] was to construct a Hadamard space for which the Izeki-Nayatani invariant is trivial.…”

Section: Introductionmentioning

confidence: 99%

Expanders with respect to Hadamard spaces and random graphs

Mendel¹

2015

Duke Math. J.

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Abstract. It is shown that there exists a sequence of 3-regular graphs {G n } ∞ n=1 and a Hadamard space X such that {G n } ∞ n=1forms an expander sequence with respect to X, yet random regular graphs are not expanders with respect to X. This answers a question of [NS11]. {G n } ∞ n=1 are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods.

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“…Kondo's goal in [Kon12] was to construct a Hadamard space for which the Izeki-Nayatani invariant is trivial. He succeeded to do so by using the Euclidean cone over certain expander graphs.…”

Section: Introductionmentioning

confidence: 99%

“…We are grateful to Uriel Feige, Konstantin Makarychev and Yuval Peres for helpful discussions, and to Jon Kleinberg for formulating Question 2.4. We also thank Takefumi Kondo for sending us a preliminary version of [Kon12]. M. …”

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confidence: 99%