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On the Rate of Convergence to the Asymptotic Cone for Nilpotent Groups and Subfinsler Geometry

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On the Rate of Convergence to the Asymptotic Cone for Nilpotent Groups and Subfinsler Geometry On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry Emmanuel Breuillard1 and Enrico Le Donne Laboratoire de Mathématiques, Université Paris Sud 11, 91405 Orsay, France Edited by Gregory A. Margulis, Yale University, New Haven, CT, and approved October 30, 2012 (received for review April 13, 2012) Addressing a question of Gromov, we give a rate in Pansu’s the- asymptotic cones and yet are not ð1; CÞ-quasi-isometric for any orem about the convergence to the asymptotic cone of a finitely constant C > 0, answering negatively a related question raised by generated nilpotent group equipped with a left-invariant word Burago and Margulis (6). 1 metric rescaled by a factor n. We obtain a convergence rate (mea- ForÀ generalÁ nilpotent groups,À Á we obtain a rate of convergence − 2 2 1 – n 3r − − sured in the Gromov Hausdorff metric) of for nilpotent in O n 3r if r > 2, and O n 2 if r = 2, where r is the nilpotency −1 groups of class r > 2 and n 2 for nilpotent groups of class 2. We class of Γ. The case r = 2 is a simple consequence of a result of also show that the latter result is sharp, and we make a connection Stoll (7), and we use it to obtain the 2 exponent for general r. between this sharpness and the presence of so-called abnormal 3r The above example on H ðZÞ × Z shows that this error term is geodesics in the asymptotic cone. As a corollary, we get an error 3 d d− 2 sharp for groups of nilpotency class 2. However, it is likely not to term of the form vol(B(n)) = cn + O(n 3r ) for the volume of Cay- r be sharp for groups of nilpotency class 3 or more, and we con- ley balls of a general nilpotent group of class . We also state 1 a number of related conjectural statements. jecture that the exponent 2 holds for all nilpotent groups and is therefore independent of r. – polynomial growth | subRiemannian geometry The rate of convergence in Gromov Hausdorff metric toward the asymptotic cone implies a rate of convergence in the volume asymptotics of Cayley balls. In particular, we obtain as a corollary n his fundamental paper on groups with polynomial growth, that, for every nilpotent group Γ with generating set S (S finite, Gromov (1) observed that Cayley graphs of finitely generated − I S = S 1 ∈ S groups with polynomial growth, when viewed from afar, admit , and 1 ), we have that À Á limits that are Lie groups endowed with a certain left-invariant −β n = d + d r ; → + ∞; [1] geodesic metric. This was a simple but basic step in his proof that jS j cSn OS n as n groups with polynomial growth are virtually nilpotent (i.e., have fi β = 2 > n a nilpotent subgroup of nite index). where r 3r for r 2 and S is the ball of radius n in the word In his thesis, Pansu (2) established that, if we start with the metric and cS > 0 is a constant. Cayley graph of a nilpotent group, then there is a unique limit. In Stoll had showed in (7) that one can take β = 1 for groups È À ÁÉ 2 other words, the sequence of metric spaces Cay Γ; 1 ρ of of nilpotency class at most 2, but nothing seemed known for n S n∈N higher-step groups, even though it is a folklore conjecture that Γ scaled down Cayley graphs of the nilpotent group with gen- β = 1 should hold for all r. We also note that the very fact that ρ – r erating set S and word metric S converges in the Gromov an error term exists at all is also a distinctive feature of nilpotent Hausdorff topology (3) toward a certain explicit left-invariant groups. Indeed, there is a class of (nondiscrete) groups of poly- subFinsler metric on a nilpotent Lie group: the Pansu limit 2 nomial growth (of the form R ⋊ θZ, where Z acts by an irrational metric on the asymptotic cone of Γ (Fig. 1). rotation with angle θ), which are solvable but not nilpotent, The goal of this note is to study the rate of convergence in ’ for which it can be shown that the volume asymptotics (here Pansu s theorem and give quantitative estimates. This question ≅ 3 θ was posed by Gromov [ref. 4, §2C remark 2C2(a)]. It requires volðBðnÞÞ cn ) admit no error term whatsoever if is chosen approximating (with explicit bounds) word metric geodesics in Γ carefully (8). with subFinsler geodesics in the Lie group that is the asymptotic In this note, we present the aforementioned results and give cone of Γ and vice versa. This problem is intimately connected some information on their proofs, although full details will be ∼ to the underlying geometry of nilpotent Lie groups endowed with given in a separate text (www.math.u-psud.fr/ breuilla/Balls.pdf) left-invariant subFinsler metrics. due to lack of space. One of our key findings is that the quality of the error term in The note is organized as follows. First, we state our main result Pansu’s theorem is related to the presence or the absence of so- about the rate of convergence to the asymptotic cone (Theorem called abnormal geodesics in the asymptotic cone. These geo- 2) and explain the strategy of the proof and its main ingredients. desics do not exist in classical Riemannian or Finsler geometry, In the second part of the paper, we present the example showing but are typical in subRiemannian or subFinsler geometry (5). We 1 the sharpness of the exponent 2 for step-2 groups (Theorem 7 and show that their presence worsens the error term in the conver- Proposition 8). We then describe some ideas in the proof of the gence to the asymptotic cone for general nilpotent groups. For sharpness result, in particular a detailed study of the geometry of example, if Γ = Zn, the free Abelian group of rank n,orif Γ = Z + subFinsler metrics on the Heisenberg group and its direct H2n+1ð Þ, the 2n 1-dimensional Heisenberg group, equip- product with R. Finally, in the last part of the note, we prove the ped with a word metric, then the asymptotic cone of Γ bears no volume estimate ½1, discuss the volume asymptotics conjecture, and abnormal geodesics, and it can be shown that the convergence to its relation to some well-known conjectures in subRiemannian the asymptotic cone is best possible, namely with a rate 1.On n geometry. the other hand, if Γ = H3ðZÞ × Z, the speed of convergence to- ward the asymptotic cone depends on the word metric, and while 1 for some generating sets the speed may be optimal, i.e., in n,we show that, for some choices of generating sets, the rate of con- Author contributions: E.B. and E.L.D. wrote the paper. −1 vergence is no faster that n 2. This result is due to the fact that The authors declare no conflict of interest. the asymptotic cone H3ðRÞ × R admits abnormal geodesics in the This article is a PNAS Direct Submission. direction of the second R factor (Figs. 2 and 3). In particular, we 1To whom correspondence should be addressed. E-mail: emmanuel.breuillard@math. exhibit two word metrics on H3ðZÞ × Z that have isometric u-psud.fr. 19220–19226 | PNAS | November 26, 2013 | vol. 110 | no. 48 www.pnas.org/cgi/doi/10.1073/pnas.1203854109 Downloaded by guest on September 25, 2021 measured using Pansu limit norm ║ · ║∞. A piecewise horizontal SPECIAL FEATURE path is by definition the concatenation of finitely many segments of one parameter subgroups of G∞ of the form fexpðtXÞgt∈½0;T for some X ∈ g=½g; g. The distance d∞ is geodesic and left-invariant by definition. We call it subFinsler, because the norm ║ · ║∞ is not a Euclidean norm but a polyhedral norm. It is not a Finsler metric however (if g is non-Abelian), because the norm is only defined on a subspace of the Lie algebra. It can be checked that this subspace g=½g; g z generates the whole Lie algebra. This implies [Chow’stheorem (5)] that every two points in G∞ can be joined by a piecewise fi fi y linear horizontal path and thus d∞ is well-de ned. We note nally x that it can be shown (10) that any left-invariant geodesic metric on a connected Lie group is a subFinsler metric for some norm on a generating subspace of the Lie algebra. We call the distance d∞ the Pansu limit metric of ðΓ; SÞ.Pansu showed in his thesis that ðG∞; d∞Þ is the asymptotic cone of ðΓ; ρ Þ. More precisely he showed that the sequence of renormal- S À Á Γ; 1 ρ Γ ; ized Cayley graphs Cay n S of converges toward ðG∞ d∞Þ in the Gromov–Hausdorff topology on pointed metric spaces (based ρ Γ fi at id). Here S is the left-invariant word metric on de ned ρ ; := ∈ N; −1 ∈ n by Sðx yÞ inffn x y S g. > := ; ; ; For any R 0, set X∞ðRÞ ðBd∞ ðid RÞ d∞Þ, where Bd∞ ðid RÞ is the closed ball of radius R in ðG∞; d∞Þ. Similarly, we set Fig. 1. The unit ball for the Pansu limit metric d3 of the Cayley graph of the À À Á Z discrete Heisenberg group H3ð Þ with standard generators (for an explicit := ; ; 1 ρ ; Xn RÞ BSðid RnÞ n S , where BSðid RnÞ is the closed ball of formula for d3 see Fine Geometry of the Heisenberg Group Equipped with Γ; ρ MATHEMATICS the Pansu Metric below).
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