1 6 Ju n 20 07 The wreath product of Z with Z has Hilbert compression exponent 23
نویسنده
چکیده
Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and two constants c1, c2 > 0 such that for all x, y ∈ G we have ‖ f (x)− f (y)‖2 ≥ c1d(x, y)− c2. Tessara [16] proved that the Hilbert compression exponent of the wreath product Z ≀Z is at least 3 , and Arzhantseva, Guba and Sapir [2] proved that it is at most 3 4 . Here we show that 2 3 is the correct value. Our proof is based on an application of K. Ball’s notion of Markov type.
منابع مشابه
ar X iv : 0 70 6 . 19 43 v 1 [ m at h . M G ] 1 3 Ju n 20 07 The wreath product of Z with Z has Hilbert compression exponent 23
We consider the wreath product Z ≀ Z, and prove that any Lipschitz function f : Z ≀ Z→ L2 satisfies lim inf dZ≀Z(x,y)→∞ ‖ f (x) − f (y)‖2 dZ≀Z(x, y)2/3 < ∞. On the other hand, as as shown by Tessera in [16], there exists a Lipschitz function g : Z ≀ Z → L2 and a real c > 0 such that ‖ f (x) − f (y)‖2 ≥ c dZ≀Z(x, y)2/3 for all x, y ∈ Z ≀ Z. Thus the Hilbert compression exponent of Z ≀ Z is exact...
متن کاملThe wreath product of Z with Z has Hilbert compression exponent 23
Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and a constant c > 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖2 ≥ cd(x, y). In [2] it was shown that the Hilbert compression exponent of the wreath ...
متن کاملar X iv : 0 70 6 . 19 43 v 3 [ m at h . M G ] 9 A ug 2 00 7 The wreath product of Z with Z has Hilbert compression exponent 23
Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all α ≥ 0 such that there exists a Lipschitz mapping f : G → L2 and a constant c > 0 such that for all x, y ∈ G we have ‖ f (x) − f (y)‖2 ≥ cd(x, y). In [2] it was shown that the Hilbert compression exponent of the wreath ...
متن کاملCompression bounds for wreath products
We show that if G and H are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath G ≀ H . We also prove an analogous result for coarse embeddings of wreath products. In the special case G = Z, H = Z ≀ Z our result implies that the Hilbert compression exponent of Z ≀ (Z ≀ Z) is at least 1/4, answering a question posed ...
متن کاملL p compression , traveling salesmen , and stable walks
We show that if H is a group of polynomial growth whose growth rate is at least quadratic then the Lp compression of the wreath product Z oH equals max { 1 p , 1 2 } . We also show that the Lp compression of Z oZ equals max { p 2p−1 , 2 3 } and the Lp compression of (Z oZ)0 (the zero section of Z oZ, equipped with the metric induced from Z o Z) equals max { p+1 2p , 3 4 } . The fact that the Hi...
متن کامل