Hostname: page-component-7bb8b95d7b-dtkg6 Total loading time: 0 Render date: 2024-09-12T21:16:09.014Z Has data issue: false hasContentIssue false
  • English
  • Français

Anisotropic principal series and generators of a free group

Published online by Cambridge University Press:  17 April 2009

Carlo Pensavalle
Carlo Pensavalle
Affiliation:
Istituto di Matematica Universitàdi Sassari Via Vienna 2 07100 SassariItaly
Rights & Permissions [Opens in a new window]

Extract

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.

In this paper we prove that the equivalence of anisotropic principal series of a free group Г related to different generator sets induces a Г-isomorphism between the related Cayley-graphs. As a consequence we obtain that a nontrivial change of generators for Г leads to inequivalent anisotropic principal series.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 2 , April 1994 , pp. 205 - 217
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Aomoto, K., ‘Spectral theory on a free group and algebraic curves’, J. Fac. Sci. Univ. Tokyo Sec. IA Math 31 (1984), 297–318.Google Scholar
[2]Bishop, C. and Steger, T., ‘Three rigidity criteria for PSL(2, R)’, Bull. Amer. Math. Soc. 24 (1991), 117–123.CrossRefGoogle Scholar
[3]Cartier, P., ‘Geometric et analyse sur le arbres’, Sem. Bourbaki 24 (1971–72).Expose 407 Lecture Notes in Math. 317 (Springer Verlag, Berlin, 1973), pp. 123–140.Google Scholar
[4]Cowling, M. and Steger, T., ‘The irreducibility of restrictions of unitary representations of lattices’, J. Reine Angew. Math. 420 (1991), 85–98.Google Scholar
[5]Culler, M. and Morgan, J., ‘Groups acting on R-trees’, Proc. London Math. Soc. 55 (1985), 571–604.Google Scholar
[6]Figà-Talamanca, A. and Nebbia, C., Harmonic analysis and representation theory for groups acting on homogeneous trees, London Math. Soc. Lecture Note Series 162 (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[7]Figà-Talamanca, A. and Picardello, A.M., Harmonic analysis on free groups, Lecture Notes in Pure and Appl. Math. 87 (Marcel Dekker, New York, 1983).Google Scholar
[8]Figà-Talamanca, A. and Steger, T., ‘Harmonic analysis on trees’, Sympos. Math. 29 (1987), 163–182.Google Scholar
[9]Gerl, P. and Woess, W., ‘Local limits and harmonic functions for nonisotropic random walks on free groups’, Probab. Theory Related Fields 71 (1986), 341–355.CrossRefGoogle Scholar
[10]Kesten, H., ‘Symmetric random walks on groups’, Trans. Amer. Math. Soc. 92 (1959), 336–354.CrossRefGoogle Scholar
[11]Kuhn, G. and Steger, T., ‘A characterization of spherical series representations of free group’, Proc. Amer. Math. Soc. 113 (1991), 1085–1096.CrossRefGoogle Scholar
[12]Mantero, A.M. and Zappa, A., ‘The Poisson transform on free groups and uniformly bounded representations’, J. Funct. Anal. 51 (1983), 372–399.CrossRefGoogle Scholar
[13]Mantero, A.M. and Zappa, A., ‘Irreducibility of the analytic continuation of the principal series of a free group’, J. Austral. Math. Soc. Ser. A 43 (1987), 199–210.CrossRefGoogle Scholar
[14]Pensavalle, C., ‘Spherical principal series and generators of a free group’, Boll. Un. Math. Ital. (to appear).Google Scholar
[15]Pytlik, T., ‘Radial functions on free groups and a decomposition of the regular representation into irreducible components’, J. Reine Angew. Math. 326 (1981), 123–135.Google Scholar
[16]Woess, W., ‘Context-free languages and random walks on groups’, Discrete Math. 67 (1987), 81–87.CrossRefGoogle Scholar