Skip to main content Accessibility help
Hostname: page-component-684899dbb8-t7hbd Total loading time: 0.428 Render date: 2022-05-20T21:02:06.655Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

14 - Buildings, Groups of Lie Type and Random Walks

Published online by Cambridge University Press:  20 July 2017

James Parkinson
School of Mathematics and Statistics, University of Sydney, Carslaw Building, F07, NSW, 2006, Australia
Tullio Ceccherini-Silberstein
Università degli Studi del Sannio, Italy
Maura Salvatori
Università degli Studi di Milano
Ecaterina Sava-Huss
Cornell University, New York
Get access


Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1] P., Abramenko K., Brown Buildings: theory and applications, Graduate Texts in Mathematics, 248, Springer, 2008.
[2] J.-P., Anker, B., Schapira, B., Trojan, Heat kernel and Green function estimates on affine building of type Ãr, preprint, 2006.
[3] L., Bartholdi, M., Neuhauser, W., Woess, Horocyclic products of trees, J. Eur. Math. Soc. 10, 771–816, 2008.Google Scholar
[4] N., Bourbaki, Lie groups and lie algebras, Chapters 4–6, Elements of mathematics, Springer-Verlag, Berlin Heidelberg New York, 2002.
[5] P., Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup., 14, no. 4, 403–32, 1981.Google Scholar
[6] M., Bridson, A., Haefliger, Metric spaces of non-positive curvature, Grundlehren 319, Springer-Verlag, 1999.
[7] K., Brown, P., Diaconis, Random walks and hyperplane arrangements, Ann. Probab. 26, 4, 1813–54, 1998.Google Scholar
[8] K., Brown, Semigroups, rings, and Markov chains, J. Theoret. Probab. 13, 871–938, 2000.Google Scholar
[9] K., Brown, Semigroup and ring theoretical methods in probability, Representations of finite dimensional algebras and related topics in Lie theory and geometry, 326, Fields Inst. Commun., 40, Amer. Math. Soc., Providence, RI, 2004.
[10] F., Bruhat, J., Tits, Groupes réductifs sur un corps local, Publ. Math. IHES, 41, 5–251, 1972.Google Scholar
[11] R., Carter, Simple Groups of Lie Type, Wiley, 1989.
[12] D. I., Cartwright, W., M_lotkowski, Harmonic analysis for groups acting on triangle buildings, Journal of the Australian Mathematical Society A, 56, 345–83, 1994.Google Scholar
[13] D. I., Cartwright, V., Kaimanovich, W., Woess, Random walks on the affine group of local fields and of homogeneous trees, Annales de l'Institut Fourier 44, no. 4, 1243–88, 1994.Google Scholar
[14] D. I., Cartwright, W., Woess, Isotropic random walks in a building of type Ãd, Mathematische Zeitschrift, 247, 101–35, 2004.Google Scholar
[15] H. S. M., Coxeter, The complete enumeration of finite groups of the form r 2 i = (ri rj)kij = 1, J. London Math. Soc. 10, 21–5, 1935.Google Scholar
[16] M. W., Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monograph Series, Princeton University Press, 32, 2008.
[17] P., Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics lecture notes, vol. 11, 1988.Google Scholar
[18] P., Diaconis, A., Ram, Analysis of systematic scan metropolis algorithms using Iwahori-Hecke algebra techniques, Michigan Math. J. 48, 2000.Google Scholar
[19] W., Feit, G., Higman, The nonexistence of certain generalized polygons, J. Algebra, 1, 114–31, 1964.Google Scholar
[20] M., Geck, G., Pfeiffer, Characters of finite Coxeter groups and Iwahori- Hecke algebras, Oxford University Press, 2000.
[21] L., Gilch, S., Müller, J., Parkinson, Limit theorems for random walks on Fuchsian buildings and Kac-Moody groups: Groups, geometry, and dynamics, to appear 2017.
[22] Y., Guivarc'h, M., Keane, P., Roynette, Marches aléatoires sur les groupes de Lie, LNM Vol. 624, Springer-Verlag, 1977.
[23] Y., Guivarc'h, Loi des grands nombres et rayon spectral d'une marche aléatoire sur un groupe de Lie, Astérisque, vol. 74, 47–98, 1980.Google Scholar
[24] A., Gyoja, K., Uno, On the semisimplicity of Hecke algebras, J. Math. Soc. Japan 41, 75–9, 1989.Google Scholar
[25] P., Haôssinski, P., Mathieu, S., Müller Renewal theory for random walks on surface groups, preprint, 2013.
[26] J. E., Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, 1990.Google Scholar
[27] N., Iwahori, H., Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups, Publications Mathématiques de l'IHÉS 25, 5–48, 1965.Google Scholar
[28] V., Kac, Infinite dimensional dimensional Lie algebras, 3rd edn, Cambridge University Press, 1990.
[29] V., Kaimanovich, Lyapunov exponents, symmetric spaces, and a multiplicative ergodic theorem for semisimple Lie groups. J. Soviet Math. 47, 2387–98, 1989.Google Scholar
[30] W., Kantor, Generalized polygons, SCABs and GABs, In: Buildings and the Geometry of Diagrams, Proceedings, Como, Lecture Notes in Mathematics, Springer-Verlag, 1181, 79–158, 1984.
[31] R., Kilmoyer, L., Solomon, On the theorem of Feit-Higman, Journal of Combinatorial Theory, Series A, 15, 3, 310–22, 1973.Google Scholar
[32] S. P., Lalley, Finite range random walk on free groups and homogeneous trees, Ann. Probab., 21 (4), 2087–130, 1993.Google Scholar
[33] M., Lindlbauer, M., Voit, Limit theorems for isotropic random walks on triangle buildings, J. Aust. Math. Soc, 73, 301–33, 2002.Google Scholar
[34] I. G., Macdonald, Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute, 2, University of Madras, 1971.
[35] E., Opdam, On the spectral decomposition of affine Hecke algebras, J. Inst. Math. Jussieu, 3, No. 4, 531–648, 2004.Google Scholar
[36] J., Parkinson, Buildings and Hecke algebras, J. Algebra, 297, 1–49, 2006.Google Scholar
[37] J., Parkinson, Spherical harmonic analysis on affine buildings, Math. Z. 253, 571–606, 2006.Google Scholar
[38] J., Parkinson, Isotropic random walks on affine buildings, Annales de l'Institut Fourier, 57, No.2, 379–419, 2007.Google Scholar
[39] J., Parkinson, B., Schapira, A local limit theorem for random walks on the chambers of à 2 buildings, Progress in Probability, 64, Birkhäuser, 15–53, 2011.
[40] J., Parkinson, On calibrated representations and the Plancherel theorem for affine Hecke algebras, J. Algebraic Combin. 40, 331–71, 2014.Google Scholar
[41] J., Parkinson, W., Woess, Random walks and regular sequences in affine buildings, Ann. Inst. Fourier, 65, 675–707, 2015.Google Scholar
[42] M., Ronan, Lectures on buildings, University of Chicago Press, 2009.
[43] M., Ronan, A construction of buildings with no rank 3 residues of spherical type. Buildings and the Geometry of Diagrams, Lecture Notes in Mathematics, 1181, 242–8, 1986.Google Scholar
[44] L., Saloff-Coste, W., Woess, Transition operators, groups, norms, and spectral radii, Pacific Journal of Mathematics, 180, no. 2, 1997.Google Scholar
[45] S., Sawyer, Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete 42, 279–92, 1978.Google Scholar
[46] B., Schapira, Random walk on a building of type Ãr and Brownian motion of the Weyl chamber, Annales de l'I.H.P. (B) 45, 289–301, 2009.Google Scholar
[47] R., Steinberg, Lecture notes on Chevalley groups, Yale University, 1967.
[48] D., Stroock, S. R. S., Varadhan, Limit theorems for random walks on Lie groups, Indian J. of Stat., Sankhy.a, Ser. A, 35, 277–94, 1973.Google Scholar
[49] J., Tits, Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, vol. 386 Springer-Verlag, Berlin-New York, 1974.
[50] J., Tits, Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Invent. Math. 43, 283–95, 1977.Google Scholar
[51] J., Tits, A local approach to buildings, The geometric vein: The Coxeter Festschrift, Springer-Verlag, 519–47, 1981.
[52] J., Tits, Immeubles de type affine, Buildings and the geometry of diagrams (Como, 1984), 159–90, Lecture Notes in Mathematics, 1181, Springer, Berlin, 1986.
[53] J., Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra, 105(2), 542–73, 1987.Google Scholar
[54] F., Tolli, A local limit theorem on certain p-adic groups and buildings, Monatsh. Math., 133, 163–73, 2001.Google Scholar
[55] B., Trojan, Heat kernel and Green function estimates on affine buildings, preprint, 2013.
[56] H. Van, Maldeghem, Generalized polygons, Monographs in Mathematics, 93, Birkhäuser, Basel, Boston, Berlin, 1998.
[57] A. D., Virtser, Central limit theorem for semi-simple Lie groups, Theory Probab. Appl. 15, 667–87, 1970.Google Scholar
[58] D., Wehn, Probabilities on Lie groups, Proc. Nat. Acad. Sci. USA, 48, 791–5, 1962.Google Scholar
[59] R., Weiss, The structure of affine buildings, Annals of Mathematics Studies, 168, Princeton University Press, 2009.
[60] W., Woess, Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics 138, Cambridge University Press, 2000.
Cited by

Save book to Kindle

To save this book to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats