Advertisement

Filling invariants of stratified nilpotent Lie groups

  • Moritz Gruber
Article
  • 7 Downloads

Abstract

Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of the Euclidean space and in the high dimensions slower than the filling functions of the Euclidean space. We do this by developing a purely algebraic condition on the Lie algebra of a stratified nilpotent Lie group. Further, we find a sufficient criterion for such groups to have a filling function in a special dimension with faster growth as the appropriate filling function of the Euclidean space. Further we bound the higher divergence functions of stratified nilpotent Lie groups.

Keywords

Filling functions Higher divergence functions Nilpotent Lie groups 

References

  1. 1.
    Abrams, A., Brady, N., Dani, P., Duchin, M., Young, R.: Pushing fillings in right-angled Artin groups. J. Lond. Math. Soc. (2) 87(3), 663–688 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Allcock, D.: An isoperimetric inequality for the Heisenberg groups. Geom. Funct. Anal. 8(2), 219–233 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosio, L., Kirchheim, B.: Currents in metric spaces. Acta Math. 185(1), 1–80 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bowditch, B.H.: A short proof that a subquadratic isoperimetric inequality implies a linear one. Mich. Math. J. 42(1), 103–107 (1995)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brady, N., Farb, B.: Filling-invariants at infinity for manifolds of nonpositive curvature. Trans. Am. Math. Soc. 350(8), 3393–3405 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Breuillard, E.: Geometry of locally compact groups of polynomial growth and shape of large balls. Groups Geom. Dyn. 8(3), 669–732 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burillo, J.: Lower bounds of isoperimetric functions for nilpotent groups. In: Geometric and computational perspectives on infinite groups (Minneapolis, MN and New Brunswick, NJ, 1994), volume 25 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pp. 1–8. Amer. Math. Soc., Providence, RI (1996)Google Scholar
  8. 8.
    Capogna, L., Danielli, D., Pauls Scott, D., Tyson, J.T.: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem volume 259 of Progress in Mathematics. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  9. 9.
    D’Ambra, G.: Induced subbundles and Nash’s implicit function theorem. Differ. Geom. Appl. 4(1), 91–105 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Eliashberg, Y., Mishachev, N.: Introduction to the \(h\)-principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2002)Google Scholar
  11. 11.
    Federer, H., Fleming, W.H.: Normal and integral currents. Ann. Math. 2(72), 458–520 (1960)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gromov, M.: Partial differential relations, volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer, Berlin (1986). [Results in mathematics and related areas (3)]Google Scholar
  13. 13.
    Gromov, M.: Carnot-Carathéodory spaces seen from within. In: sub-riemannian geometry, volume 144 of Progr. Math., pp. 79–323. Birkhäuser, Basel (1996)Google Scholar
  14. 14.
    Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces, volume 152 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA (1999). Based on the 1981 French original [ MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael BatesGoogle Scholar
  15. 15.
    Gruber, M.: The growth of the first non-Euclidean filling function of the quaternionic Heisenberg Group. arXiv:1702.00954, to appear in Advances in Geometry
  16. 16.
    Gruber, M.: Large Scale Geometry of Stratified Nilpotent Lie Groups. DissertationGoogle Scholar
  17. 17.
    Korányi, A., Ricci, F.: A classification-free construction of rank-one symmetric spaces. Bull. Kerala Math. Assoc. (Spec. Issue) 2005, 73–88 (2007)Google Scholar
  18. 18.
    Enrico L., Séverine R.: Besicovitch covering property on graded groups and applications to measure differentiation. arXiv:1512.04936
  19. 19.
    Leuzinger, E.: Corank and asymptotic filling-invariants for symmetric spaces. Geom. Funct. Anal. 10(4), 863–873 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Leuzinger, E.: Optimal higher-dimensional Dehn functions for some CAT(0) lattices. Groups Geom. Dyn. 8(2), 441–466 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Leuzinger, E., Pittet, C.: On quadratic Dehn functions. Math. Z. 248(4), 725–755 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Magnani, V.: Unrectifiability and rigidity in stratified groups. Arch. Math. (Basel) 83(6), 568–576 (2004)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mitchell, J.: On Carnot-Carathéodory metrics. J. Differ. Geom. 21(1), 35–45 (1985)CrossRefGoogle Scholar
  25. 25.
    Niblo, G.A., Roller, M.A. (eds.): Geometric group theory. Vol. 2, volume 182 of London mathematical society lecture note series. Cambridge University Press, Cambridge (1993)Google Scholar
  26. 26.
    Olshanskii, A.Y., Sapir, M.V.: Quadratic isometric functions of the Heisenberg groups. A combinatorial proof. J. Math. Sci. (N.Y.) 93(6), 921–927 (1999). Algebra, 11MathSciNetCrossRefGoogle Scholar
  27. 27.
    Pansu, P.: Croissance des boules et des géodésiques fermées dans les nilvariétés. Ergod. Theory Dyn. Syst. 3(3), 415–445 (1983)CrossRefGoogle Scholar
  28. 28.
    Pittet, Christophe: Isoperimetric inequalities for homogeneous nilpotent groups. In: Geometric group theory (Columbus, OH, 1992), volume 3 of Ohio State Univ. Math. Res. Inst. Publ., pages 159–164. de Gruyter, Berlin (1995)Google Scholar
  29. 29.
    Pittet, C.: Isoperimetric inequalities in nilpotent groups. J. Lond. Math. Soc. (2) 55(3), 588–600 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Raghunathan, M.S.: Discrete subgroups of Lie groups. Springer, New York, Heidelberg (1972). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68CrossRefGoogle Scholar
  31. 31.
    Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and geometry on groups, volume 100 of Cambridge tracts in mathematics. Cambridge University Press, Cambridge (1992)Google Scholar
  32. 32.
    Wenger, S.: Isoperimetric inequalities of Euclidean type in metric spaces. Geom. Funct. Anal. 15(2), 534–554 (2005)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wenger, S.: Filling invariants at infinity and the Euclidean rank of Hadamard spaces. Int. Math. Res. Not. 33, 83090 (2006)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Wenger, S.: Nilpotent groups without exactly polynomial Dehn function. J. Topol. 4(1), 141–160 (2011)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Wolf, J.A.: Curvature in nilpotent Lie groups. Proc. Am. Math. Soc. 15, 271–274 (1964)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Young, R.: Homological and homotopical higher-order filling functions. Groups Geom. Dyn. 5(3), 683–690 (2011)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Young, R.: Filling inequalities for nilpotent groups through approximations. Groups Geom. Dyn. 7(4), 977–1011 (2013)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Young, R.: High-dimensional fillings in Heisenberg groups. J. Geom. Anal. 26(2), 1596–1616 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations