Strong Embeddability for Groups Acting on Metric Spaces

Abstract

The strong embeddability is a notion of metric geometry, which is an intermediate property lying between coarse embeddability and property A. In this paper, the permanence properties of strong embeddability for groups acting on metric spaces are studied. The authors show that a finitely generated group acting on a finitely asymptotic dimension metric space by isometries whose K-stabilizers are strongly embeddable is strongly embeddable. Moreover, they prove that the fundamental group of a graph of groups with strongly embeddable vertex groups is also strongly embeddable.

Keywords

Strong embeddability / Groups action / Graph of groups / Relative hyperbolic groups

References

[1] Arzhantseva G., Tessera R.. Relative expanders. Geom. Funct. Anal., 2015, 25: 317-341

[2] Bell G.. Property A for groups acting on metric spaces. Topol. Appl., 2003, 130: 239-251

[3] Bell G., Dranishnikov A.. On asymptotic dimension of groups. Algebr. Geom. Topol., 2001, 1: 57-71

[4] Bell G., Dranishnikov A.. On asymptotic dimension of groups acting on trees. Geom. Dedicata., 2004, 103: 89-101

[5] Bell G., Dranishnikov A.. Asymptotic dimension. Topol. Appl., 2008, 55: 1265-1296

[6] Chen X., Wang X.. Operator norm localization property of relative hyperbolic group and graph of groups. J. Funct. Anal., 2008, 255: 642-656

[7] Dadarlat M., Guentner E.. Uniform embeddability of relatively hyperbolic groups. J. Reine Angew. Math., 2007, 612: 1-15

[8] Ferry S., Ranicki A., Rosenberg J.. Novikov Conjectures, Index Theorems and Rigidity, 1995, Cambridge University Press, Cambridge: London Mathematical Society Lecture Notes

[9] Gromov M.. Niblo A., Roller M.. Asymptotic invariants of infinite groups, Geometric Group Theory. London Mathematical Society Lecture Notes, 1993, Cambridge: Cambridge University Press 1-295

[10] Higson N., Roe J.. Amenable group actions and the Novikov Conjecture. J. Reine Angew. Math., 2000, 519: 143-153

[11] Ji R., Ogle C., Ramsey B. W.. Relative property A and relative amenability for countable groups. Adv. Math., 2012, 231: 2734-2754

[12] Ji, R., Ogle, C. and Ramsey, B. W., Strong embeddablility and extensions of groups, arXiv: 1307.1935.

[13] Osin D.. Asymptotic dimension of relatively hyperbolic groups. Int. Math. Res. Notices., 2005, 35: 2143-2161

[14] Sako H.. Property A and the operator norm localization property for discrete metric spaces. J. Reine Angew. Math., 2014, 690: 207-216

[15] Serre J.-P.. Trees, 1980, Berlin Heidelberg, New York: Springer-Verlag

[16] Tu J.-L.. Remarks on Yu’s property A for disrete metric spaces and groups. Bull. Soc. Math. France, 2001, 129(1): 115-139

[17] Willett R.. Some notes on property A, Limits of Graphs Group Theory and Computer Science, 2009 191-281

[18] Xia J., Wang X.. On strong embeddability and finite decomposition complexity. Acta Math. Sinica (English Series), 2017, 33(3): 403-418

[19] Yu G.. The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math., 2000, 139(1): 201-240