Abstract
In this paper, the author studies the coarse embedding into uniformly convex Banach spaces. The author proves that the property of coarse embedding into Banach spaces can be preserved under taking the union of the metric spaces under certain conditions. As an application, for a group G strongly relatively hyperbolic to a subgroup H, the author proves that B(n) = {g ∈ G | |g| S∪ℋ ≤ n} admits a coarse embedding into a uniformly convex Banach space if H does.
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References
Benyamini, Y. and Lindenstrauss, J., Geometric Nonlinear Functional Analysis, Vol. 48, American Mathematical Society, Providence, RI, 2000.
Brown, N. and Guentner, E., Uniform embeddings of bounded geometry spaces into reflexive Banach space, Proc. Amer. Math. Soc., 133, 2005, 2045–2050.
Chen, X., Wang, Q. and Yu, G., The maximal coarse Baum-Connes conjecture for spaces which admit a fibred coarse embedding into Hilbert space, arXiv.org/abs/1208.4543v1, 2012.
Dadarlat, M. and Guentner, E., Constructions preserving Hilbert space uniform embeddability of discrete groups, Trans. Amer. Math. Soc., 355, 2003, 3253–3275.
Dadarlat, M. and Guentner, E., Uniform embeddability of relatively hyperbolic groups, J. Reine Angew. Math., 612, 2007, 1–15.
Farb, B., Relatively hyperbolic groups, Geom. Funct. Anal., 8(5), 1998, 810–840.
Ferry, S. C., Ranicki, A. and Rosenberg, J., A History Survey of the Novikov Conjecture, Novikov Conjectures, Index Theorems and Rigidity, London Mathematical Society Lecture Notes, Vol. 1, London Math. Soc. Lecture Note, Ser. 226, Cambridge University Press, Cambridge, 1995.
Fukaya, T. and Oguni, S., The coarse Baum-Connes conjecture for relatively hyperbolic groups, J. Topol. Anal., 4(1), 2012, 99–113.
Gromov, M., Asymptotic Invariants of Infinite Groups, Geometric Group Theory, Vol. 2, London Math. Soc. Lecture Notes, Ser. 182, Cambridge University Press, Cambridge, 1993.
Johnson, W. B. and Lindenstrauss, J., Handbook of geometry of Banach spaces, Elsevier, Amsterdam, New York, 2001.
Johnson, W. B. and Randrianarivony, N. L., ℓ p (p > 2) does not coarsely embed into a Hilbert space, Proc. Amer. Math. Soc., 134(4), 2006, 1045–1050.
Kasparov, G. and Yu, G., The coarse geometric Novikov conjecture and uniform convexity, Adv. Math., 206(1), 2006, 1–56.
Lafforgue, V., Un renforcement de la properiété, Duke Math. J., 143(3), 2008, 559–602.
Osin, D., Asymptotic dimension of relatively hyperbolic groups, Int. Math. Res. Not., 35, 2005, 2143–2161.
Roe, J., Lectures on Coarse Geometry, University Lecture Series, Vol. 31, Amer. Math. Soc., Providence, RI, 2003.
Yu, G., The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., 139(1), 2000, 201–240.
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This work was supported by the National Natural Science Foundation of China (No. 11301566) and the Postdoc Scholarship (No. 2012M511900).
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Ren, Q. Coarse embedding into uniformly convex Banach spaces. Chin. Ann. Math. Ser. B 35, 733–742 (2014). https://doi.org/10.1007/s11401-014-0855-7
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DOI: https://doi.org/10.1007/s11401-014-0855-7