Skip to main content Accessibility help
×
Home

LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS

  • BERNHARD KRÖN (a1), JÖRG LEHNERT (a2), NORBERT SEIFTER (a3) and ELMAR TEUFL (a4)

Abstract

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.

Copyright

References

Hide All
1.Alexopoulos, G. K., Random walks on discrete groups of polynomial volume growth, Ann. Probab. 30 (2) (2002), 723801, doi:10.1214/aop/1023481007, MR1905856 (2003d:60010), Zbl1023.60007.
2.Babai, L., Some applications of graph contractions, J. Graph Theory 1 (2) (1977), 125130, Special issue dedicated to Paul Turán, doi:10.1002/jgt.3190010207, MR0460171 (57 #167), Zbl0381.05029.
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.
4.Bourbaki, N., Elements of mathematics. General topology. Part 1, (Hermann, Paris, 1966), MR0205210 (34 #5044a), Zbl0301.54001.
5.Breuillard, E., Geometry of locally compact groups of polynomial growth and shape of large balls, preprint, 2012.
6.Paul Bonnington, C., Richter, R. Bruce and Watkins, Mark E., Between ends and fibers, J. Graph Theory 54 (2) (2007), 125153, doi:10.1002/jgt.20202, MR2285455 (2007k:05100), Zbl1118.05017.
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.
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.
9.Eberlein, P. and O'Neill, B., Visibility manifolds, Pacific J. Math. 46 (1973), 45109, doi:10.2140/pjm.1973.46.45, MR0336648 (49 #1421), Zbl0264.53026.
10.Freudenthal, H., Neuaufbau der Endentheorie, Ann. of Math. 43 (2) (1942), 261–279, doi:10.2307/1968869, MR0006504 (3,315a), Zbl0060.40006.
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), 333338, doi:10.1007/BF01788688, MR1032384 (91c:05092), Zbl0714.05029.
12.Gluškov, V. M., Locally nilpotent locally bicompact groups Trudy Moskov. Mat. Obšč. 4 (1955), 291332, MR0072422 (17,281b), Zbl0068.01901.
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.
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), 183213, MR624814 (82m:53035), Zbl0467.53035.
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), 1295, MR1253544 (95m:20041), Zbl0841.20039.
16.Guivarc'h, Y., Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333379, MR0369608 (51 #5841), Zbl0294.43003, http://www.numdam.org/item?id=BSMF_1973__101__333_0.
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), 4798, 3, MR588157 (82g:60016), Zbl0448.60007.
18.Hochschild, G. P., The structure of Lie groups (Holden-Day Inc., San Francisco, 1965), MR0207883 (34 #7696), Zbl0131.02702.
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.
20.Jung, H. A. and Niemeyer, P., Decomposing ends of locally finite graphs Math. Nachr. 174 (1995), 185202, doi:10.1002/mana.19951740113, MR1349044 (96h:05120), Zbl0833.05052.
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), 477484, MR1224725 (94d:05075), Zbl0787.05048.
22.Kaimanovich, V. A., Poisson boundaries of random walks on discrete solvable groups, Probability measures on groups, X (Oberwolfach, 1990) (Plenum, New York, 1991), 205238, MR1178986 (94m:60014), Zbl0823.60006.
23.Krön, B., Lehnert, J. and Stein, M. J., Linear boundary and HNN-extensions, preprint, 2014.
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.
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.
26.Pansu, P., Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math. (2) 129 (1) (1989), 160, doi:10.2307/1971484, MR979599 (90e:53058), Zbl0678.53042.
27.Sabidussi, G., Vertex-transitive graphs Monatsh. Math. 68 (1964), 426438, doi:10.1007/BF01304186, MR0175815 (31 #91), Zbl0136.44608.
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.
29.Seifter, N., Groups acting on graphs with polynomial growth, Discrete Math. 89 (3) (1991), 269280, doi:10.1016/0012-365X(91)90120-Q, MR1112445 (92g:05099), Zbl0739.05041.
30.Seifter, N., Properties of graphs with polynomial growth, J. Combin. Theory Ser. B 52 (2) (1991), 222235, doi:10.1016/0095-8956(91)90064-Q, MR1110471 (92i:05106), Zbl0668.05034.
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.
32.Tanaka, R., Large deviation on a covering graph with group of polynomial growth, Math. Z. 267 (3–4) (2011), 803833, doi:10.1007/s00209-009-0647-z, MR2776059 (2012i:60051).
33.Tanaka, R., 2012, private communication.
34.Trofimov, V. I., Bounded automorphisms of graphs and the characterization of grids, Izv. Akad. Nauk SSSR Ser. Mat. 47 (2) (1983), 407420, MR697303 (84h:05067), Zbl0519.05037.
35.Trofimov, V. I., Graphs with polynomial growth, Mat. Sb. (N.S.) 123 (165)(3) (1984), 407421, doi:10.1070/SM1985v051n02ABEH002866, MR735714 (85m:05041), Zbl0548.05033.
36.Willard, S., General topology (Dover Publications Inc., Mineola, NY, 2004), Reprint of the 1970 original [Addison-Wesley, Reading, MA; MR0264581], MR2048350, Zbl1052.54001.

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed