SOME REMARKS ON GENERALIZED ROUNDNESS GHISLAIN JAUDON
arXiv:0704.3371v2 [math.GR] 14 May 2007
Abstract. By using the links between generalized round...

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arXiv:0704.3371v2 [math.GR] 14 May 2007

Abstract. By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank ≥ 2 equals 1. This answers a question of J-F. Lafont and S. Prassidis.

1. Introduction Generalized roundness (see definition below) was introduced by P. Enflo in [E69a] and [E69b] in order to study the uniform structure of metric spaces, and as an application of this notion he gave a solution to Smirnov’s problem [E69b]. Rudiments of a general theory for generalized roundness were developed in [LTW97], where the link of this notion with negative type inequalities is emphasized. More recently, generalized roundness was investigated in the case of finitely generated groups [LP06]. Unfortunately, generalized roundness is very difficult to estimate in general, and for this reason there are only very few examples of metric spaces for which the exact value is known. Here we use ideas developed in [LTW97] together with estimates on generalized roundness computed in [LP06] and results about equivariant Hilbert space compression in [GK04] to deduce exact values of the generalized roundness of finitely generated free (abelian and non-abelian) groups endowed with their standard metrics. 2. Preliminaries Let (X, d) be a metric space, and let G denote a group acting on X by isometries. 2.1. Definition. The generalized roundness of (X, d) is the supremum of all positive numbers p such that for every n ≥ 2 and any collection of 2n points {a1 . . . , an , b1 , . . . , bn } in X, the following inequality holds: X X (d(ai , aj )p + d(bi , bj )p ) ≤ d(ai , bj )p . 1≤i

1≤i,j≤n

We will denote the generalized roundness of the metric space (X, d) by gr(X, d), and simply gr(X) when there is no ambiguity about the metric d. Essentially, a metric space (X, d) satisfies gr(X, d) = p if 2n-gons (for every n ≥ 2) are thinner than the ones in Lp -spaces. This observation is justified by the following result (see [LTW97]): 2.2. Proposition. Let 1 ≤ p ≤ 2 and (X, B, µ) be a measured space. Then gr(Lp (X, B, µ)) = p. 2.3. Remark. The generalized roundness of any infinite and finitely generated group (endowed with the word metric) is always ≤ 2 (see [LP06] Proposition 4.7). 2.4. Definition. A function ψ : X × X → R is said to be a kernel of negative type if ψ(x, x) = 0 for all x ∈ X, ψ(x, y) = ψ(y, x) for all x, Pyn ∈ X, and if for every integer n ≥ 1, for every x1 , . . . , xn ∈ X and for every λ1 , . . . , λn ∈ R satisfying i=1 λi = 0, the following inequality holds: X λi λj ψ(xi , xj ) ≤ 0. 1≤i,j≤n

The kernel is said to be G-invariant if ψ(gx, gy) = ψ(x, y) for all x, y ∈ X and for all g ∈ G. Date: February 1, 2008. 2000 Mathematics Subject Classification. Primary 51F99, Secondary 20F65. Key words and phrases. Generalized roundness, negative type functions, Hilbert space compression, CAT(0) cube complexes. This work was supported by the Swiss National Science Foundation Grant ♯ PP002-68627. 1

Kernels of negative type and generalized roundness are related by the following result (see [LTW97]): 2.5. Theorem. gr(X, d) ≥ p if and only if dp is a kernel of negative type. 2.6. Definition. Let H be an Hilbert space. A map f : X → H is said to be a uniform embedding of X into H if there exist non-decreasing functions ρ± (f ) : R+ → R+ such that: (i) (ii)

ρ− (f )(d(x, y)) ≤ kf (x) − f (y)kH ≤ ρ+ (f )(d(x, y)), for all x, y ∈ X; limr→+∞ ρ± (f )(r) = +∞.

Then the G-equivariant Hilbert space compression of the metric space X, denoted by RG (X), is defined as the supremum of all 0 < β ≤ 1 for which there exists a G-equivariant uniform embedding f into some Hilbert space which is equipped with an action of G by affine isometries, such that ρ+ (f ) is affine and ρ− (f )(r) = rβ (for large enough r). Concerning negative definite kernels, we will need a G-invariant analogue of the so-called GNS-construction (see for instance [BHV06], 2.10): 2.7. Proposition. Let ψ be a G-invariant kernel of negative type on X, then there exists a Hilbert space H equipped with an action of G by affine isometries, and a G-equivariant map f : X → H, such that ψ(x, y) = kf (x) − f (y)k2H for all x, y ∈ X. Theorem 2.5 combined with Proposition 2.7 immediately gives the following estimate: 2.8. Proposition. For every group G of isometries of X, RG (X) ≥

gr(X) 2 .

2.9. Remarks. On one hand, the previous inequality cannot be improved. Indeed, let X = G = Z acting on itself by left translations and being endowed with its usual left invariant word metric. Considering the inclusion of Z into R, which is a Z-equivariant isometry, we have RZ (Z) = 1 and moreover gr(Z) ≥ gr(R). But Proposition 2.2 gives gr(R) = 2. Therefore, by Remark 2.3, we obtain that gr(Z) = 2 = 2RZ (Z). On the other hand, the inequality is unfortunately not an equality in general. Consider for instance the case X = G = Z2 . The Hilbert space compression of Z2 equals 1 (see [GK04] Example 2.7), and by amenability the equivariant Hilbert space compression of Z2 equals the Hilbert space compression (see [CTV06] Proposition 4.4). Hence RZ2 (Z2 ) = 1. But, by Corollary 3.2 below, gr(Z2 ) = 1. 3. Negative type inequalities in CAT(0) cube complexes Recall that a cube complex is a metric polyhedral complex in which each cell is isometric to an Euclidean cube [− 12 , 21 ]n , and the gluing maps are isometries. A finite dimensional cube complex always carries a complete geodesic metric (see [BH99]). A cube complex is CAT(0) if it is simply connected and if, in the link of every cube of the complex, there is at most one edge between any two vertices and there is no triangle not contained in a 2-simplex (see [BH99] and [G87]). Let X denote a finite dimensional CAT(0) cube complex. The 0-skeleton X (0) of X can be endowed with the metric, denoted by d0 , given by the length of the shortest edge path in the 1-skeleton of X between vertices. The proof of the next result is strongly inspired by [CN04] Example 1. 3.1. Theorem. Let X be a finite dimensional CAT(0) cube complex. Then gr(X (0) , d0 ) ≥ 1. Proof. By Proposition 2.2, it is sufficient to exhibit an isometric embedding of (X (0) , d0 ) into some L1 -space. Given an edge in the complex, there is a unique isometrically embedded codimension 1 coordinate hyperplane (again called hyperplane) which cuts this edge transversely in its midpoint, and this hyperplane separates the complex into two components, called half spaces (see [G87]). We will denote by H the set of all hyperplanes. Moreover, by [S95], shortest edge paths in the 1-skeleton cross any hyperplane at most once. Hence the distance between two vertices, d0 (v, w), is the number of hyperplanes separating v and w (hyperplanes such that the two vertices are not in the same half space). We fix a vertex v0 ∈ X (0) and for every vertex v ∈ X (0) we set Hv := {h ∈ H | h separates v0 and v}. Then we define X δh f : X (0) → l1 (H), v 7→ h∈Hv

2

where

(

1 if k = h 0 otherwise It remains to show that f is an isometry. Let v, w be two vertices of X. Then X X X kf (v) − f (w)kl1 (H) = δh (l) − δh (l) . δh : H → R, k 7→

l∈H h∈Hv

(∗)

h∈Hw

For every l ∈ H, we have

( X X 1 δh (l) = δh (l) − 0 h∈Hv

h∈Hw

if l ∈ Hv △Hw if l ∈ Hv ∩ Hw

But a hyperplane l ∈ H separates v and w if and only if l ∈ Hv △Hw . Hence, the sum in the left member of (∗) is exactly the number of hyperplanes separating v and w, i.e., kf (v) − f (w)kl1 (H) = d0 (v, w). 3.2. Corollary. Let n ≥ 2. We endow Zn with the word metric associated to its canonical basis, and we endow the free group of rank n, Fn , with the word metric associated to any free generating system. We have: (i) gr(Zn ) = 1; (ii) gr(Fn ) = 1. Proof. (i). By Corollary 4.14 of [LP06], we have gr(Zn ) ≤ 1. For the converse inequality, let us consider the action of Zn on Rn by left translations. Rn can be viewed naturally as a CAT(0) cube complex X of which the 0-skeleton (endowed with the metric d0 ) is isometric to Zn . Therefore, Theorem 3.1 gives the result. (ii). The Cayley graph of Fn is a tree. In particular, this is a 1-dimensional CAT(0) cube complex. Hence, by Theorem 3.1, we obtain that gr(Fn ) ≥ 1. On the other hand, it is known that RFn (Fn ) = 21 (see [GK04]). Then by Proposition 2.8, we deduce that gr(Fn ) ≤ 1. 3.3. Remark. Let G be a group acting freely by isometries on the 0-skeleton (X (0) , d0 ) of a CAT(0) cube complex X. If we fix a vertex v0 , we define a metric D0 on G by setting D0 (g, h) := d0 (gv0 , hv0 ), and Theorem 3.1 gives gr(G, D0 ) ≥ 1. References [BH99] M.R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften Volume 319 [Fundamental principles of mathematical Sciences], Springer-Verlag, Berlin, 1999 [BHV06] M.B. Bekka, P. de la Harpe, A. Valette, Kazhdan’s property (T), (to appear) [CN04] S. Campbell, G. Niblo, Hilbert space compression and exactness for discrete groups, Journal of Funct. Anal. 222, p. 292-305 (2004) [CTV06] Y. Cornulier, R. Tessera, A. Valette, Isometric group actions on Hilbert spaces: growth of cocycles, preprint (2006), to appear in GAFA [E69a] P. Enflo, On the non-existence of uniform homeomorphisms between Lp -spaces, Ark. Mat. 8 (1969), p. 103-105 [E69b] P. Enflo, On a problem of Smirnov, Ark. Mat. 8 (1969), p. 107-109 [G87] M. Gromov, Hyperbolic groups, “Esssays in group theory”, S.M. Gersten (ed.), MSRI Publ. 8, Springer-Verlag (1987), p. 75-267 [GK04] E. Guentner, J. Kaminker, Exactness and uniform embeddability of discrete groups, J. London Math. Soc. (2) 70, n3, p. 703-718 (2004) [LTW97] C.J. Lennard, A. M. Tonge, A. Weston, Generalized roundness and negative type, Michigan Math. J. 44 (1997), p. 37-45 [LP06] J-F. Lafont, S. Prassidis, Roundness properties of groups, Geom. Dedicata 117 (2006), p. 137-160 [S95] M. Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), p. 585-617 ´matiques, 2-4 rue du Li` Universit´ e de Gen` eve, Section de Mathe evre, Case postale 64, 1211 Gen` eve 4, Switzerland E-mail address: [email protected]

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