Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T11:00:55.734Z Has data issue: false hasContentIssue false
  • English
  • Français

LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

Part of:Special aspects of infinite or finite groupsLie groupsSpaces with richer structuresGraph theory

Published online by Cambridge University Press:  22 December 2014

BERNHARD KRÖN,
JÖRG LEHNERT,
NORBERT SEIFTER and
ELMAR TEUFL
BERNHARD KRÖN
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Vienna, Austria e-mail: bernhard.kroen@univie.ac.at
JÖRG LEHNERT
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany e-mail: lehnert@mis.mpg.de
NORBERT SEIFTER
Affiliation:
Department Mathematik und Informationstechnologie, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700 Leoben, Austria e-mail: seifter@unileoben.ac.at
ELMAR TEUFL
Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: elmar.teufl@uni-tuebingen.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We define a pseudometric on the set of all unbounded subsets of a metric space. The Kolmogorov quotient of this pseudometric space is a complete metric space. The definition of the pseudometric is guided by the principle that two unbounded subsets have distance 0 whenever they stay sublinearly close. Based on this pseudometric we introduce and study a general concept of boundaries of metric spaces. Such a boundary is the closure of a subset in the Kolmogorov quotient determined by an arbitrarily chosen family of unbounded subsets. Our interest lies in those boundaries which we get by choosing unbounded cyclic sub(semi)groups of a finitely generated group (or more general of a compactly generated, locally compact Hausdorff group). We show that these boundaries are quasi-isometric invariants and determine them in the case of nilpotent groups as a disjoint union of certain spheres (or projective spaces). In addition we apply this concept to vertex-transitive graphs with polynomial growth and to random walks on nilpotent groups.

MSC classification

Primary:20F65: Geometric group theory
Secondary:54E35: Metric spaces, metrizability20F18: Nilpotent groups22E25: Nilpotent and solvable Lie groups05C63: Infinite graphs

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015, pp. 591 - 632
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Alexopoulos, G. K., Random walks on discrete groups of polynomial volume growth, Ann. Probab. 30 (2) (2002), 723–801, doi:10.1214/aop/1023481007, MR1905856 (2003d:60010), Zbl1023.60007.CrossRefGoogle Scholar
2.Babai, L., Some applications of graph contractions, J. Graph Theory 1 (2) (1977), 125–130, Special issue dedicated to Paul Turán, doi:10.1002/jgt.3190010207, MR0460171 (57 #167), Zbl0381.05029.CrossRefGoogle Scholar
3.Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319 (Springer-Verlag, Berlin, 1999), MR1744486 (2000k:53038), Zbl0988.53001.CrossRefGoogle Scholar
4.Bourbaki, N., Elements of mathematics. General topology. Part 1, (Hermann, Paris, 1966), MR0205210 (34 #5044a), Zbl0301.54001.Google Scholar
5.Breuillard, E., Geometry of locally compact groups of polynomial growth and shape of large balls, preprint, 2012.Google Scholar
6.Paul Bonnington, C., Richter, R. Bruce and Watkins, Mark E., Between ends and fibers, J. Graph Theory 54 (2) (2007), 125–153, doi:10.1002/jgt.20202, MR2285455 (2007k:05100), Zbl1118.05017.CrossRefGoogle Scholar
7.Corwin, L. J. and Greenleaf, F. P., Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics, vol. 18 (Cambridge University Press, Cambridge, 1990), Basic theory and examples, MR1070979 (92b:22007), Zbl0704.22007.Google Scholar
8.de la Harpe, P., Topics in geometric group theory, Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 2000), MR1786869 (2001i:20081), Zbl0965.20025.Google Scholar
9.Eberlein, P. and O'Neill, B., Visibility manifolds, Pacific J. Math. 46 (1973), 45–109, doi:10.2140/pjm.1973.46.45, MR0336648 (49 #1421), Zbl0264.53026.CrossRefGoogle Scholar
10.Freudenthal, H., Neuaufbau der Endentheorie, Ann. of Math. 43 (2) (1942), 261–279, doi:10.2307/1968869, MR0006504 (3,315a), Zbl0060.40006.CrossRefGoogle Scholar
11.Godsil, C. D., Imrich, W., Seifter, N., Watkins, M. E. and Woess, W., A note on bounded automorphisms of infinite graphs, Graphs Combin. 5 (4) (1989), 333–338, doi:10.1007/BF01788688, MR1032384 (91c:05092), Zbl0714.05029.CrossRefGoogle Scholar
12.Gluškov, V. M., Locally nilpotent locally bicompact groups Trudy Moskov. Mat. Obšč. 4 (1955), 291–332, MR0072422 (17,281b), Zbl0068.01901.Google Scholar
13.Goodman, R. W., Nilpotent Lie groups: structure and applications to analysis, Lecture Notes in Mathematics, vol. 562 (Springer-Verlag, Berlin, 1976), MR0442149 (56 #537), Zbl0347.22001.CrossRefGoogle Scholar
14.Gromov, M., Hyperbolic manifolds, groups and actions, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Ann. of Math. Stud., vol. 97 (Princeton University Press, Princeton, N.J., 1981), 183–213, MR624814 (82m:53035), Zbl0467.53035.Google Scholar
15.Gromov, M., Asymptotic invariants of infinite groups, Geometric group theory, vol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182 (Cambridge University Press, Cambridge, 1993), 1–295, MR1253544 (95m:20041), Zbl0841.20039.Google Scholar
16.Guivarc'h, Y., Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333–379, MR0369608 (51 #5841), Zbl0294.43003, http://www.numdam.org/item?id=BSMF_1973__101__333_0.CrossRefGoogle Scholar
17.Guivarc'h, Y., Sur la loi des grands nombres et le rayon spectral d'une marche aléatoire, Conference on Random Walks (Kleebach, 1979) (French), Astérisque, vol. 74 (Soc. Math. France, Paris, 1980), 47–98, 3, MR588157 (82g:60016), Zbl0448.60007.Google Scholar
18.Hochschild, G. P., The structure of Lie groups (Holden-Day Inc., San Francisco, 1965), MR0207883 (34 #7696), Zbl0131.02702.Google Scholar
19.Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115 (Springer-Verlag, Berlin, 1979), Structure of topological groups, integration theory, group representations, MR551496 (81k:43001), Zbl0416.43001.CrossRefGoogle Scholar
20.Jung, H. A. and Niemeyer, P., Decomposing ends of locally finite graphs Math. Nachr. 174 (1995), 185–202, doi:10.1002/mana.19951740113, MR1349044 (96h:05120), Zbl0833.05052.CrossRefGoogle Scholar
21.Jung, H. A., Notes on rays and automorphisms of locally finite graphs, Graph structure theory (Seattle, WA, 1991), Contemporary Mathematics, vol. 147 (American Mathematical Society, Providence, RI, 1993), 477–484, MR1224725 (94d:05075), Zbl0787.05048.Google Scholar
22.Kaimanovich, V. A., Poisson boundaries of random walks on discrete solvable groups, Probability measures on groups, X (Oberwolfach, 1990) (Plenum, New York, 1991), 205–238, MR1178986 (94m:60014), Zbl0823.60006.Google Scholar
23.Krön, B., Lehnert, J. and Stein, M. J., Linear boundary and HNN-extensions, preprint, 2014.Google Scholar
24.Mal'tsev, A. I., On a class of homogeneous spaces, Amer. Math. Soc. Transl. 1951 (39) (1951), 33, MR0039734 (12,589e), ZblZbl 0034.01701.Google Scholar
25.Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory, 2nd ed. (Dover Publications Inc., Mineola, NY, 2004), Presentations of groups in terms of generators and relations, MR2109550 (2005h:20052), Zbl1130.20307.Google Scholar
26.Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math. (2) 129 (1) (1989), 1–60, doi:10.2307/1971484, MR979599 (90e:53058), Zbl0678.53042.CrossRefGoogle Scholar
27.Sabidussi, G., Vertex-transitive graphs Monatsh. Math. 68 (1964), 426–438, doi:10.1007/BF01304186, MR0175815 (31 #91), Zbl0136.44608.CrossRefGoogle Scholar
28.Segal, D., Polycyclic groups, Cambridge Tracts in Mathematics, vol. 82 (Cambridge University Press, Cambridge, 1983), doi:10.1017/CBO9780511565953, MR713786 (85h:20003), Zbl0516.20001.CrossRefGoogle Scholar
29.Seifter, N., Groups acting on graphs with polynomial growth, Discrete Math. 89 (3) (1991), 269–280, doi:10.1016/0012-365X(91)90120-Q, MR1112445 (92g:05099), Zbl0739.05041.CrossRefGoogle Scholar
30.Seifter, N., Properties of graphs with polynomial growth, J. Combin. Theory Ser. B 52 (2) (1991), 222–235, doi:10.1016/0095-8956(91)90064-Q, MR1110471 (92i:05106), Zbl0668.05034.CrossRefGoogle Scholar
31.Stroppel, M., Locally compact groups, EMS Textbooks in Mathematics (European Mathematical Society (EMS), Zürich, 2006), doi:10.4171/016, MR2226087 (2007d:22001), Zbl1102.22005.CrossRefGoogle Scholar
32.Tanaka, R., Large deviation on a covering graph with group of polynomial growth, Math. Z. 267 (3–4) (2011), 803–833, doi:10.1007/s00209-009-0647-z, MR2776059 (2012i:60051).CrossRefGoogle Scholar
33.Tanaka, R., 2012, private communication.Google Scholar
34.Trofimov, V. I., Bounded automorphisms of graphs and the characterization of grids, Izv. Akad. Nauk SSSR Ser. Mat. 47 (2) (1983), 407–420, MR697303 (84h:05067), Zbl0519.05037.Google Scholar
35.Trofimov, V. I., Graphs with polynomial growth, Mat. Sb. (N.S.) 123 (165)(3) (1984), 407–421, doi:10.1070/SM1985v051n02ABEH002866, MR735714 (85m:05041), Zbl0548.05033.Google Scholar
36.Willard, S., General topology (Dover Publications Inc., Mineola, NY, 2004), Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581], MR2048350, Zbl1052.54001.Google Scholar