We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first -cohomology, or the existence of proper isometric actions on whose orbits are almost quasi-isometries.
Nous introduisons différentes notions de profil isopérimétrique à grande échelle d’un groupe localement compact, compactement engendré, et moyennable. Ces quantités asymptotiques permettent de mesurer la moyennabilité du groupe. En particulier, nous nous intéressons à la classe des groupes moyennables à croissance exponentielle qui en ce sens sont les “plus moyennables possibles". Nous montrons que ces groupes partagent divers propriétés intéressantes, comme la vitesse de décroissance de la probabilité de retour des marches aléatoires, l’annulation de leur cohomologie- réduite en degré , ou bien l’existence d’actions par isométries affines sur dont les orbites sont presque des quasi-isométries.
@article{TSG_2006-2007__25__179_0, author = {Tessera, Romain}, title = {Large-scale isoperimetry on locally compact groups and applications}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {179--188}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {25}, year = {2006-2007}, doi = {10.5802/tsg.255}, zbl = {1167.22004}, mrnumber = {2478816}, language = {en}, url = {http://www.numdam.org/articles/10.5802/tsg.255/} }
TY - JOUR AU - Tessera, Romain TI - Large-scale isoperimetry on locally compact groups and applications JO - Séminaire de théorie spectrale et géométrie PY - 2006-2007 SP - 179 EP - 188 VL - 25 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.255/ DO - 10.5802/tsg.255 LA - en ID - TSG_2006-2007__25__179_0 ER -
%0 Journal Article %A Tessera, Romain %T Large-scale isoperimetry on locally compact groups and applications %J Séminaire de théorie spectrale et géométrie %D 2006-2007 %P 179-188 %V 25 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.255/ %R 10.5802/tsg.255 %G en %F TSG_2006-2007__25__179_0
Tessera, Romain. Large-scale isoperimetry on locally compact groups and applications. Séminaire de théorie spectrale et géométrie, Volume 25 (2006-2007), pp. 179-188. doi : 10.5802/tsg.255. http://www.numdam.org/articles/10.5802/tsg.255/
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