Isoperimetry of nilpotent groups

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.

Key words： Nilpotent groups; nilpotent Lie groups; Dehn functions; filling functions

References

1 Alonso J M, Wang X, Pride S J. Higher-dimensional isoperimetric (or Dehn) functions of groups. J Group Theory, 1999, 2(1): 81–112

2 Bridson M R. On the geometry of normal forms in discrete groups. Proc Lond Math Soc (3), 1993, 67(3): 596–616

3 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

4 Burillo J, Taback J. Equivalence of geometric and combinatorial Dehn functions. New York J Math, 2002, 8: 169–179 (electronic)

5 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

6 Federer H, Fleming W H. Normal and integral currents. Ann of Math (2), 1960, 72: 458–520

7 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

8 Gromov M. Filling Riemannian manifolds. J Differential Geom, 1983, 18(1): 1–147

9 Gromov M. Metric Structures for Riemannian and Non-Riemannian Spaces. Progr Math, Vol 152. Boston: Birkhäuser, 1999

10 Gruber M. Large Scale Geometry of Stratified Nilpotent Lie Groups (in preparation)

11 Korányi A, Ricci F. A classification-free construction of rank-one symmetric spaces. Bull Kerala Math Assoc, 2005, (Special Issue): 73–88 (2007)

12 Leuzinger E. Optimal higher-dimensional Dehn functions for some CAT(0) lattices. Groups Geom Dyn, 2014, 8(2): 441–466

13 Leuzinger E, Pittet C. On quadratic Dehn functions. Math Z, 2004, 248(4): 725–755

14 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

15 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

16 Pittet C. Isoperimetric inequalities in nilpotent groups. J Lond Math Soc (2), 1997, 55(3): 588–600

17 Raghunathan M S. Discrete Subgroups of Lie Groups. Ergeb Math Grenzgeb, Band 68. New York-Heidelberg: Springer-Verlag, 1972

18 Varopoulos N Th, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge Tracts in Math, Vol 100. Cambridge: Cambridge Univ Press, 1992

19 Wenger S. A short proof of Gromov’s filling inequality. Proc Amer Math Soc, 2008, 136(8): 2937–2941

20 Wenger S. Nilpotent groups without exactly polynomial Dehn function. J Topol, 2011, 4(1): 141–160

21 Wolf J A. Curvature in nilpotent Lie groups. Proc Amer Math Soc, 1964, 15: 271–274

22 Young R. High-dimensional fillings in Heisenberg groups. 2010, arXiv: 1006.1636v2

23 Young R. Homological and homotopical higher-order filling functions. Groups Geom Dyn, 2011, 5(3): 683–690

24 Young R. Filling inequalities for nilpotent groups through approximations. Groups Geom Dyn, 2013, 7(4): 977–1011