Abstract
This survey gives an overview of the isoperimetric properties of nilpotent groups and Lie groups. It discusses results for Dehn functions and filling functions as well as the techniques used to retrieve them. The content reaches from long standing results up to the most recent development.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alonso J M, Wang X, Pride S J. Higher-dimensional isoperimetric (or Dehn) functions of groups. J Group Theory, 1999, 2(1): 81–112
Bridson M R. On the geometry of normal forms in discrete groups. Proc Lond Math Soc (3), 1993, 67(3): 596–616
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). DIMACS Ser Discrete Math Theoret Comput Sci, Vol 25. Providence: Amer Math Soc, 1996, 1–8
Burillo J, Taback J. Equivalence of geometric and combinatorial Dehn functions. New York J Math, 2002, 8: 169–179 (electronic)
Epstein D B A, Cannon J W, Holt D F, Levy S V F, Paterson M S, Thurston W P. Word Processing in Groups. Boston: Jones and Bartlett Publishers, 1992
Federer H, Fleming W H. Normal and integral currents. Ann of Math (2), 1960, 72: 458–520
Gersten S M. Isoperimetric and isodiametric functions of finite presentations. In: Geometric Group Theory, Vol 1 (Sussex, 1991). London Math Soc Lecture Note Ser, Vol 181. Cambridge: Cambridge Univ Press, 1993, 79–96
Gromov M. Filling Riemannian manifolds. J Differential Geom, 1983, 18(1): 1–147
Gromov M. Metric Structures for Riemannian and Non-Riemannian Spaces. Progr Math, Vol 152. Boston: Birkhäuser, 1999
Gruber M. Large Scale Geometry of Stratified Nilpotent Lie Groups (in preparation)
Korányi A, Ricci F. A classification-free construction of rank-one symmetric spaces. Bull Kerala Math Assoc, 2005, (Special Issue): 73–88 (2007)
Leuzinger E. Optimal higher-dimensional Dehn functions for some CAT(0) lattices. Groups Geom Dyn, 2014, 8(2): 441–466
Leuzinger E, Pittet C. On quadratic Dehn functions. Math Z, 2004, 248(4): 725–755
Niblo G A, Roller M A, eds. Geometric Group Theory, Vol 2. London Math Soc Lecture Note Ser, Vol 182. Cambridge: Cambridge Univ Press, 1993
Pittet C. Isoperimetric inequalities for homogeneous nilpotent groups. In: Geometric Group Theory (Columbus, OH, 1992). Ohio State Univ Math Res Inst Publ, Vol 3. Berlin: de Gruyter, 1995, 159–164
Pittet C. Isoperimetric inequalities in nilpotent groups. J Lond Math Soc (2), 1997, 55(3): 588–600
Raghunathan M S. Discrete Subgroups of Lie Groups. Ergeb Math Grenzgeb, Band 68. New York-Heidelberg: Springer-Verlag, 1972
Varopoulos N Th, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge Tracts in Math, Vol 100. Cambridge: Cambridge Univ Press, 1992
Wenger S. A short proof of Gromov’s filling inequality. Proc Amer Math Soc, 2008, 136(8): 2937–2941
Wenger S. Nilpotent groups without exactly polynomial Dehn function. J Topol, 2011, 4(1): 141–160
Wolf J A. Curvature in nilpotent Lie groups. Proc Amer Math Soc, 1964, 15: 271–274
Young R. High-dimensional fillings in Heisenberg groups. 2010, arXiv: 1006.1636v2
Young R. Homological and homotopical higher-order filling functions. Groups Geom Dyn, 2011, 5(3): 683–690
Young R. Filling inequalities for nilpotent groups through approximations. Groups Geom Dyn, 2013, 7(4): 977–1011
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gruber, M. Isoperimetry of nilpotent groups. Front. Math. China 11, 1239–1258 (2016). https://doi.org/10.1007/s11464-016-0577-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-016-0577-0