Some remarks on generalized roundness
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 r…
Authors: Ghislain Jaudon
SOME REMARKS ON GENER ALIZED R OUNDNESS GHISLAIN JAUDON Abstract. By using the links b et wee n generalized roundness, negativ e type inequalities and equiv ariant Hilb ert space compressions, we obt ain that the generalized roundness of the usua l Ca yley graph of finitely generated free groups and fr ee abelian groups of r ank ≥ 2 equals 1. This answers a question of J-F. Lafont and S. Prassidis. 1. Introduction Generalized roundness (see definition b elow) w as in tro duced by P . Enflo in [E69a] and [E69b] in order to study the uniform str ucture of metric spaces, and as a n a pplica tion o f this notion he gave a so lution to Smir nov’s problem [E6 9b]. Rudimen ts of a gener al theor y for generalized r oundness were develop ed in [L TW9 7], where the link o f this notion with nega tive t yp e ineq ualities is emphasized. More recently , generalized r oundness was in vestigated in the case of finitely g enerated gro ups [LP06]. Unfortunately , generalized ro undness is very difficult to estimate in gener al, and for this rea son there a re only v ery few examples of metric spaces for which the ex act v alue is known. Here we use ideas dev elop ed in [L TW97] together with estima tes on generalized roundness computed in [LP06] and results abo ut equiv ariant Hilber t space compression in [GK04] to deduce exact v alues of the generalized roundness of finitely generated fr e e (abelia n and non- ab elian) gr oups endow ed with their standa rd metric s . 2. Preliminaries Let ( X, d ) be a metric space, and let G deno te a group acting on X by isometries . 2.1. Defini tion. The gener alize d r oundness of ( X , d ) is the supremum of all p ositive num ber s p such that for every n ≥ 2 and any collection of 2 n p oints { a 1 . . . , a n , b 1 , . . . , b n } in X , the following inequality holds: X 1 ≤ i
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