Duke Mathematical Journal

A quasi-isometric embedding theorem for groups

Alexander Yu. Olshanskii and Denis V. Osin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that every group H of at most exponential growth with respect to some left invariant metric admits a bi-Lipschitz embedding into a finitely generated group G such that G is amenable (resp., solvable, satisfies a nontrivial identity, elementary amenable, of finite decomposition complexity) whenever H also shares those conditions. We also discuss some applications to compression functions of Lipschitz embeddings into uniformly convex Banach spaces, Følner functions, and elementary classes of amenable groups.

Article information

Source
Duke Math. J., Volume 162, Number 9 (2013), 1621-1648.

Dates
First available in Project Euclid: 11 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1370955541

Digital Object Identifier
doi:10.1215/00127094-2266251

Mathematical Reviews number (MathSciNet)
MR3079257

Zentralblatt MATH identifier
1331.20054

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F69: Asymptotic properties of groups
Secondary: 20F16: Solvable groups, supersolvable groups [See also 20D10] 20E22: Extensions, wreath products, and other compositions [See also 20J05]

Citation

Olshanskii, Alexander Yu.; Osin, Denis V. A quasi-isometric embedding theorem for groups. Duke Math. J. 162 (2013), no. 9, 1621--1648. doi:10.1215/00127094-2266251. https://projecteuclid.org/euclid.dmj/1370955541


Export citation

References

  • [1] G. Arzhantseva, C. Drutu, and M. Sapir, Compression functions of uniform embeddings of groups into Hilbert and Banach spaces, J. Reine Angew. Math. 633 (2009), 213–235.
  • [2] G. Arzhantseva, V. Guba, and M. Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv. 81 (2006), 911–929.
  • [3] T. Austin, Amenable groups with very poor compression into Lebesgue spaces, Duke Math. J. 159 (2011), 187–222.
  • [4] M. G. Brin, Elementary amenable subgroups of R. Thompson’s group $F$, Internat. J. Algebra Comput. 15 (2005), 619–642.
  • [5] N. Brown, E. Guentner, Uniform embeddings of bounded geometry spaces into reflexive Banach spaces, Proc. Amer. Math. Soc. 133 (2005), 2045–2050.
  • [6] C. Chou, Elementary amenable groups, Illinois J. Math. 24 (1980), 396–407.
  • [7] M. Dadarlat and E. Guentner, Constructions preserving Hilbert space uniform embeddability of discrete groups, Trans. Amer. Math. Soc. 355, no. 8 (2003), 3253–3275.
  • [8] T. C. Davis and A. Yu. Olshanskii, Relative subgroup growth and subgroup distortion, arXiv:1212.5208v1 [math.GR].
  • [9] A. Erschler, On isoperimetric profiles of finitely generated groups, Geom. Dedicata 100 (2003), 157–171.
  • [10] A. Erschler, Piecewise automatic groups, Duke Math. J. 134 (2006), 591–613.
  • [11] M. Gromov, “Asymptotic invariants of infinite groups” in Geometric Group Theory, Vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, 1991, 1–295.
  • [12] M. Gromov, Entropy and isoperimetry for linear and non-linear group actions, Groups Geom. Dyn. 2 (2008), 499–593.
  • [13] E. Guentner and J. Kaminker, Exactness and uniform embeddability of discrete groups, J. Lond. Math. Soc. (2) 70 (2004), 703–718.
  • [14] E. Guentner, R. Tessera, and G. Yu, A notion of geometric complexity and its application to topological rigidity, Invent. Math. 189 (2012), 315–357.
  • [15] P. Hall, “On the embedding of a group in a join of given groups” in Collection of Articles Dedicated to the Memory of Hanna Neumann, VIII, J. Aust. Math. Soc. 17, Jerusalem Academic Press, Jerusalem, 1974, 434–495.
  • [16] G. Higman, B. H. Neumann, and H. Neumann, Embedding theorems for groups, J. Lond. Math. Soc. 24 (1949), 247–254.
  • [17] G. Kasparov and G. Yu, The coarse geometric Novikov conjecture and uniform convexity, Adv. Math. 206 (2006), 1–56.
  • [18] V. Lafforgue, Un renforcement de la propriété (T), Duke Math. J. 143 (2008), 559–602.
  • [19] W. Magnus, On a theorem of Marshall Hall, Ann. of Math. (2) 40 (1939), 764–768.
  • [20] B. H. Neumann and H. Neumann, Embedding theorems for groups, J. Lond. Math. Soc. 34 (1959), 465–479.
  • [21] A. Yu. Olshanskii, “Distortion functions for subgroups” in Geometric Group Theory Down Under (Canberra, 1996), Walter de Gruyter, Berlin, 1999, 281–291.
  • [22] R. E. Phillips, Embedding methods for periodic groups, Proc. Lond. Math. Soc. (3) 35 (1977), 238–256.
  • [23] D. A. Ramras, R. Tessera, G. Yu, Finite decomposition complexity and the integral Novikov conjecture for higher algebraic K-theory, preprint, arXiv:1111.7022v4 [math.KT].
  • [24] V. N. Remeslennikov and V. G. Sokolov, Certain properties of the Magnus imbedding (in Russian), Algebra Logika 9 (1970), 566–578; English translation in Algebra Logic 9 (1970), 342–349.
  • [25] R. Tessera, Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces, Comment. Math. Helv. 86 (2011), 499–535.
  • [26] A. Vershik, Amenability and approximation of infinite groups: Selected translations. Selecta Math. (N.S.) 2 (1982), no. 4, 311–330.
  • [27] J. von Neumann, Zur allgemeinen Theorie des Masses, Fund. Math. 13 (1929), 73–116.
  • [28] J. S. Wilson, Embedding theorems for residually finite groups, Math. Z. 174 (1980), 149–157.
  • [29] G. Yu, The Coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert spaces, Invent. Math. 139 (2000), 201–240.