# Large-scale isoperimetry on locally compact groups and applications

Romain Tessera^{[1]}

- [1] Vanderbilt University Department of mathematics StevensonCenter, Nashville, TN 37240 United

Séminaire de théorie spectrale et géométrie (2006-2007)

- Volume: 25, page 179-188
- ISSN: 1624-5458

## Access Full Article

top## Abstract

top## How to cite

topTessera, Romain. "Large-scale isoperimetry on locally compact groups and applications." Séminaire de théorie spectrale et géométrie 25 (2006-2007): 179-188. <http://eudml.org/doc/11223>.

@article{Tessera2006-2007,

abstract = {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 $L^p$-cohomology, or the existence of proper isometric actions on $L^p$ whose orbits are almost quasi-isometries.},

affiliation = {Vanderbilt University Department of mathematics StevensonCenter, Nashville, TN 37240 United},

author = {Tessera, Romain},

journal = {Séminaire de théorie spectrale et géométrie},

keywords = {amenable; -isoperimetric profile; elementary solvable group; geometrically elementary solvable group; random walk; -cohomology},

language = {eng},

pages = {179-188},

publisher = {Institut Fourier},

title = {Large-scale isoperimetry on locally compact groups and applications},

url = {http://eudml.org/doc/11223},

volume = {25},

year = {2006-2007},

}

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

PB - Institut Fourier

VL - 25

SP - 179

EP - 188

AB - 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 $L^p$-cohomology, or the existence of proper isometric actions on $L^p$ whose orbits are almost quasi-isometries.

LA - eng

KW - amenable; -isoperimetric profile; elementary solvable group; geometrically elementary solvable group; random walk; -cohomology

UR - http://eudml.org/doc/11223

ER -

## References

top- T. Coulhon. (2000). Random walks and geometry on infinite graphs. Lecture notes on analysis on metric spaces, Luigi Ambrosio, Francesco Serra Cassano, eds., 5-30. Zbl1063.60063MR2023121
- P.A. Cherix, M. Cowling, P. Jolissaint, P. Julg, A. Valette. Groups with the Haagerup Property. Birkhäuser, Progress in Mathematics 197, 2001. Zbl1030.43002MR1852148
- T. Coulhon and A. Grigor’yan, A geometric approach to on-diagonal heat kernel lower bounds on groups Contemp. Math. , A.M.S. (2004) 65-99., Ann. Inst. Fourier, 51, 6 (2001) 1763-1827. Zbl1137.58307
- A. Cheeger, M. Gromov. ${L}^{2}$-cohomology and group cohomology. Topology 25, 189-215, 1986. Zbl0597.57020MR837621
- T. Coulhon et L. Saloff-Coste. (1995). Variétés riemanniennes isométriques à l’infini. Rev. Mat. Iberoamericana 11, 3, 687-726. Zbl0845.58054
- T. Coulhon et L. Saloff-Coste. (1993).Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9, 2, 293-314. Zbl0782.53066MR1232845
- Y. de Cornulier, R. Tessera, Alain Valette. Isometric group actions on Hilbert spaces: growth of cocycles. To appear in GAFA. Zbl1129.22004
- A. Erschler. (2003). On isoperimetric profiles of finitely generated groups. Geom. dedicata 100, 157-171. Zbl1049.20024MR2011120
- A. Grigor’yan. (1994). Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana 10, 395-452. Zbl0810.58040
- P. Hall. The Frattini subgroups of finitely generated groups. Proc. London Math. Soc. (3), 11: 327-352, 1961. Zbl0104.02201MR124406
- E. Kappos. ${\ell}^{p}$-cohomology for groups of type $F{P}_{n}$. math.FA/0511002, 2006.
- S. Mustapha. Distorsion des distances dans les groupes $p$-adiques. Bulletin des Sciences Mathématiques. 124, Issue 3 , 175-191, 2000. Zbl0948.22012MR1753262
- S. Mustapha. Sami Bornes inférieures pour les marches aléatoires sur les groupes $p$-adiques moyennables. Ann. Inst. H. Poincaré Probab. Statist. 42, no. 1, 81–88, 2006. Zbl1102.60041MR2196972
- P. Pansu. Cohomologie ${L}^{p}$ des variétés à courbure négative, cas du degré 1. Rend. Semin. Mat., Torino Fasc. Spec., 95-120, 1989. Zbl0723.53023MR1086210
- C. Pittet. (2000). The isoperimetric profile of homogeneous Riemannian manifolds. J. Diff. Geom. 54, 2, 255-302. Zbl1035.53069MR1818180
- C. Pittet, L. Saloff-Coste. (2003). Random walks on finite rank solvable groups. J. Eur. Math. Soc. 5, 313-342. Zbl1057.20026MR2017850
- R. Tessera. (2006). Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. math.GR/0603138. Zbl1274.43009
- R. Tessera. (2006). Vanishing of the first reduced cohomology with values in a ${L}^{p}$-representation. math.GT/0611001.
- R. Tessera. (2007). Large scale Sobolev inequalities on metric measure spaces and applications. Zbl1194.53036
- R. Tessera. (2007). Isoperimetric profile of subgroups and probability of return of random walks on elementary solvable groups.
- N. Varopoulos. (1988). Analysis on Lie groups. J. Funct. Anal. 76, 346-410. Zbl0634.22008MR924464

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.