Isoperimetry of nilpotent groups

doi:10.1007/s11464-016-0577-0

Frontiers of Mathematics in China

2016, Vol. 11

Issue (5): 1239-1258 https://doi.org/10.1007/s11464-016-0577-0

本期目录

Isoperimetry of nilpotent groups

Moritz GRUBER(

)

Karlsruhe Institute of Technology, Karlsruhe, Germany

全文: PDF(230 KB)

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

收稿日期: 2016-02-05 出版日期: 2016-09-23

Corresponding Author(s): Moritz GRUBER

引用本文:

. [J]. Frontiers of Mathematics in China, 2016, 11(5): 1239-1258.
Moritz GRUBER. Isoperimetry of nilpotent groups. Front. Math. China, 2016, 11(5): 1239-1258.

链接本文:

https://academic.hep.com.cn/fmc/CN/10.1007/s11464-016-0577-0
https://academic.hep.com.cn/fmc/CN/Y2016/V11/I5/1239

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 https://doi.org/10.1017/cbo9780511661860.008
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 https://doi.org/10.4171/GGD/233
13	Leuzinger E, Pittet C. On quadratic Dehn functions. Math Z, 2004, 248(4): 725–755 https://doi.org/10.1007/s00209-004-0678-4
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 https://doi.org/10.1515/9783110810820.159
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 https://doi.org/10.1007/978-3-642-86426-1
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 https://doi.org/10.1090/S0002-9939-08-09203-4
20	Wenger S. Nilpotent groups without exactly polynomial Dehn function. J Topol, 2011, 4(1): 141–160 https://doi.org/10.1112/jtopol/jtq038
21	Wolf J A. Curvature in nilpotent Lie groups. Proc Amer Math Soc, 1964, 15: 271–274 https://doi.org/10.1090/S0002-9939-1964-0162206-7
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 https://doi.org/10.4171/GGD/144
24	Young R. Filling inequalities for nilpotent groups through approximations. Groups Geom Dyn, 2013, 7(4): 977–1011 https://doi.org/10.4171/GGD/213

Viewed

Full text

Abstract

Cited

Shared

Discussed