On Hibert's Metric for Simplices

Harpe, Pierre de la

doi:10.1017/cbo9780511661860.009

Cited by 73 publications

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“…He constructed a special class of examples, now called Hilbert geometries [Hilbert 1895;, which have since attracted much interest; see, for example, [Nasu 1961;de la Harpe 1993;Karlsson and Noskov 2002;Socié-Méthou 2004;Foertsch and Karlsson 2005;Benoist 2006; Colbois and Vernicos 2007], and the two complementary surveys [Benoist 2008] and [Vernicos 2005].…”

Section: Introductionmentioning

confidence: 99%

Volume entropy of Hilbert geometries

Berck¹,

Bernig²,

Vernicos³

2010

Pacific J. Math.

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We show that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, we show that the volume entropy is bounded above by 2/(3 − d) ≤ 1, where d is the Minkowski dimension of the extremal set of K , and we construct an explicit example of a plane Hilbert geometry with noninteger volume entropy. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, we construct a new projective invariant of convex bodies, similar to the centroaffine area.

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Section: Introductionmentioning

confidence: 99%

Volume entropy of Hilbert geometries

Berck¹,

Bernig²,

Vernicos³

2010

Pacific J. Math.

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“…Proposition 3.11 yields that the group G = {t −1 ht : h ∈σ} is an infinite commutative subgroup of SL(2, Z). It is a known fact (see [4,5]) that SL(2, Z) is a word hyperbolic group, and word hyperbolic groups do not contain copies of Z ⊕ Z. Hence, G has rank one or zero.…”

Section: Proposition 311 Lat( P Ar(γ 0 )) Is Invariant As a Set Ofmentioning

confidence: 99%

Subgroups of 𝑃𝑆𝐿(3,ℂ) with four lines in general position in its limit set

Barrera¹,

Cano

Navarrete³

2011

Conform. Geom. Dyn.

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“…for some N . We remark here that if ω is stabilizing sequence, then Grigorchuk group G ω has an intermediate growth (see [8,20,27]). …”

Section: Growth Of Groups and Percolationmentioning

confidence: 99%

“…Let G ω be a group of transformations of the interval ∆ generated by a, b ω , c ω , d ω . This family of groups was introduced and analyzed by Grigorchuk in [8] (see also [20] for further references). We refer to G ω as Grigorchuk groups.…”

Section: Grigorchuk Groupmentioning

confidence: 99%

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Percolation on Grigorchuk Groups

Muchnik¹,

Pak²

2001

Communications in Algebra

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Let p c (G) be the critical probability of the site percolation on the Cayley graph of group G. In [2] of Benjamini and Schramm conjectured thatp c < 1, given the group is infinite and not a finite extension of Z. The conjecture was proved earlier for groups of polynomial and exponential growth and remains open for groups of intermediate growth.In this note we prove the conjecture for a special class of Grigorchuk groups, which is a special class of groups of intermediate growth. The proof is based on an algebraic construction. No previous knowledge of percolation is assumed.

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On Hibert's Metric for Simplices

Cited by 73 publications

References 0 publications

Volume entropy of Hilbert geometries

Volume entropy of Hilbert geometries

Subgroups of 𝑃𝑆𝐿(3,ℂ) with four lines in general position in its limit set

Percolation on Grigorchuk Groups

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