Hilbert space compression under direct limits and certain group extensions

Dreesen, Dennis

doi:10.1090/S0002-9939-2012-11341-3

Hilbert space compression under direct limits and certain group extensions
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by Dennis Dreesen

Proc. Amer. Math. Soc. 141 (2013), 421-436

DOI: https://doi.org/10.1090/S0002-9939-2012-11341-3

Published electronically: June 7, 2012

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Abstract:

We find bounds on the Hilbert space compression exponent of the limit of a directed metric system of groups. We also give estimates on the Hilbert space compression exponent of a group extension of a group $H$ by a word-hyperbolic group or a group of polynomial growth.

References

G. N. Arzhantseva, V. S. Guba, and M. V. Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv. 81 (2006), no. 4, 911–929. MR 2271228, DOI 10.4171/CMH/80
N. Brodskiy and D. Sonkin, Compression of uniform embeddings into Hilbert space, Topology Appl. 155 (2008), no. 7, 725–732. MR 2395586, DOI 10.1016/j.topol.2007.12.012
Marius Dadarlat and Erik Guentner, Constructions preserving Hilbert space uniform embeddability of discrete groups, Trans. Amer. Math. Soc. 355 (2003), no. 8, 3253–3275. MR 1974686, DOI 10.1090/S0002-9947-03-03284-7
S. M. Gersten, Bounded cocycles and combings of groups, Internat. J. Algebra Comput. 2 (1992), no. 3, 307–326. MR 1189238, DOI 10.1142/S0218196792000190
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg (eds.), Novikov conjectures, index theorems and rigidity. Vol. 1, London Mathematical Society Lecture Note Series, vol. 226, Cambridge University Press, Cambridge, 1995. Including papers from the conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, September 6–10, 1993. MR 1388294
Erik Guentner and Jerome Kaminker, Exactness and uniform embeddability of discrete groups, J. London Math. Soc. (2) 70 (2004), no. 3, 703–718. MR 2160829, DOI 10.1112/S0024610704005897
Sean Li, Compression bounds for wreath products, Proc. Amer. Math. Soc. 138 (2010), no. 8, 2701–2714. MR 2644886, DOI 10.1090/S0002-9939-10-10307-4
Assaf Naor and Yuval Peres, Embeddings of discrete groups and the speed of random walks, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn 076, 34. MR 2439557, DOI 10.1093/imrn/rnn076
Assaf Naor and Yuval Peres, $L_p$ compression, traveling salesmen, and stable walks, Duke Math. J. 157 (2011), no. 1, 53–108. MR 2783928, DOI 10.1215/00127094-2011-002
Walter D. Neumann and Lawrence Reeves, Central extensions of word hyperbolic groups, Ann. of Math. (2) 145 (1997), no. 1, 183–192. MR 1432040, DOI 10.2307/2951827
Walter D. Neumann and Lawrence Reeves, Regular cocycles and biautomatic structures, Internat. J. Algebra Comput. 6 (1996), no. 3, 313–324. MR 1404809, DOI 10.1142/S0218196796000167
G. Skandalis, J. L. Tu, and G. Yu, The coarse Baum-Connes conjecture and groupoids, Topology 41 (2002), no. 4, 807–834. MR 1905840, DOI 10.1016/S0040-9383(01)00004-0
Romain Tessera, Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv. 86 (2011), no. 3, 499–535. MR 2803851, DOI 10.4171/CMH/232
Jean-Louis Tu, Remarks on Yu’s “property A” for discrete metric spaces and groups, Bull. Soc. Math. France 129 (2001), no. 1, 115–139 (English, with English and French summaries). MR 1871980, DOI 10.24033/bsmf.2388
Guoliang Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240. MR 1728880, DOI 10.1007/s002229900032