Croissance des boules et des géodésiques fermées dans les nilvariétés

Pansu, Pierre

doi:10.1017/s0143385700002054

Cited by 173 publications

(229 citation statements)

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“…This extends the main result of Pansu [27]. The integer d(G) coincides with the exponent of growth of a naturally associated graded nilpotent Lie group, the asymptotic cone of G, and is given by the Bass-Guivarc'h formula (4) below.…”

Section: Contentssupporting

confidence: 75%

“…This was Pansu's description in [27]. However when S is not nilpotent, and is equipped with a word metric ρ Ω on a co-compact subgroup, then the determination of the limit shape, i.e.…”

Section: Asymptotic Shapesmentioning

confidence: 92%

“…In Section 5, we assume that G is a simply connected solvable Lie group and reduce the problem to the nilpotent case. In Section 6, we assume that G is a simply connected nilpotent Lie group and prove Theorem 1.4 in this case following the strategy used by Pansu in [27]. In Section 7, we prove Theorem 1.2 for general locally compact groups and reduce the proof of the results of the introduction to the Lie case.…”

Section: 4mentioning

confidence: 99%

“…Finally, let us mention that the results and methods of this paper were largely inspired by the works of Y. Guivarc'h [21] and P. Pansu [27]. .…”

Section: 4mentioning

confidence: 99%

“…See also Tits' appendix to Gromov's paper [17]. In fact Guivarc'h's Theorem 2.7 seems to have been rediscovered several times in the past 40 years, including by Pansu in his thesis [27], the latest example of that being [22].…”

Section: Definition 22 (Homogeneous Quasi-norm)mentioning

confidence: 99%

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Geometry of locally compact groups of polynomial growth and shape of large balls

Breuillard¹

2014

Groups Geom. Dyn.

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Abstract. We show that any locally compact group G with polynomial growth is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We then study the shape of large balls and show, generalizing work of P. Pansu, that after a suitable renormalization, they converge to a limiting compact set, which is isometric to the unit ball for a left-invariant subFinsler metric on the so-called graded nilshadow of S. As by-products, we obtain asymptotics for the volume of large balls, we prove that balls are Folner and hence that the ergodic theorem holds for all ball averages. Along the way we also answer negatively a question of Burago and Margulis [7] on asymptotic word metrics and recover some results of Stoll [33] of the rationality of growth series of Heisenberg groups. 1. Introduction 1.1. Groups with polynomial growth. Let G be a locally compact group with left Haar measure vol G . We will assume that G is generated by a compact symmetric subset Ω. Classically, G is said to have polynomial growth if there exist C > 0 and k > 0 such that for any integer n ≥ 1 ContentsDate: April 2012. where Ω n = Ω · . . . · Ω is the n-fold product set. Another choice for Ω would only change the constant C, but not the polynomial nature of the bound. One of the consequences of the analysis carried out in this paper is the following theorem:Theorem 1.1 (Volume asymptotics). Let G be a locally compact group with polynomial growth and Ω a compact symmetric generating subset of G. Then there exists c(Ω) > 0 and an integer d(G) ≥ 0 depending on G only such that the following holds:This extends the main result of Pansu [27]. The integer d(G) coincides with the exponent of growth of a naturally associated graded nilpotent Lie group, the asymptotic cone of G, and is given by the Bass-Guivarc'h formula (4) below. The constant c(Ω) will be interpreted as the volume of the unit ball of a subRiemannian Finsler metric on this nilpotent Lie group. Theorem 1.1 is a byproduct of our study of the asymptotic behavior of periodic pseudodistances on G, that is pseudodistances that are invariant under a co-compact subgroup of G and satisfy a weak kind of the existence of geodesics axiom (see Definition 4.1).Our first task is to get a better understanding of the structure of locally compact groups of polynomial growth. Guivarc'h [21] proved that locally compact groups of polynomial growth are amenable and unimodular and that every compactly generated 1 closed subgroup also has polynomial growth. Guivarc'h [21] and Jenkins [15] also characterized connected Lie groups with polynomial growth: a connected Lie group has polynomial growth if and only if it is of type (R), that is if for all x ∈ Lie(S), ad(x) has only purely imaginary eigenvalues. Such groups are solvable-by-compact and any connected nilpotent Lie group is of type (R).It is much more difficult to characterize discrete groups with polynomial growth, and this was done in a celebrated paper of Gromov [17], proving that they are virtually nilpotent. Losert [24] gen...

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Section: Contentssupporting

confidence: 75%

“…This was Pansu's description in [27]. However when S is not nilpotent, and is equipped with a word metric ρ Ω on a co-compact subgroup, then the determination of the limit shape, i.e.…”

Section: Asymptotic Shapesmentioning

confidence: 92%

Section: 4mentioning

confidence: 99%

“…Finally, let us mention that the results and methods of this paper were largely inspired by the works of Y. Guivarc'h [21] and P. Pansu [27]. .…”

Section: 4mentioning

confidence: 99%