×

zbMATH — the first resource for mathematics

The absolute of finitely generated groups. II: The Laplacian and degenerate parts. (English. Russian original) Zbl 07000556
Funct. Anal. Appl. 52, No. 3, 163-177 (2018); translation from Funkts. Anal. Prilozh. 52, No. 3, 3-21 (2018).
Summary: The article continues a series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into two parts, Laplacian and degenerate, and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under taking certain central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelianization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group).

MSC:
20F65 Geometric group theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Vershik, A. M., The problem of describing central measures on the path spaces of graded graphs, Funkts. Anal. Prilozhen., 48, 26-46, (2014)
[2] Vershik, A. M., Equipped graded graphs, projective limits of simplices, and their boundaries, Zap. Nauchn. Sem. POMI, 432, 83-104, (2015) · Zbl 1323.05127
[3] Vershik, A. M.; Malyutin, A. V., Phase transition in the exit boundary problem for random walks on groups, Funkts. Anal. Prilozhen., 49, 7-20, (2015) · Zbl 1328.60117
[4] Vershik, A. M., The theory of filtrations of subalgebras, standardness, and independence, Uspekhi Mat. Nauk, 72, 67-146, (2017) · Zbl 1370.05220
[5] Vershik, A. M.; Malyutin, A. V., Infinite geodesics in the discrete Heisenberg group, Zap. Nauchn. Sem. POMI, 462, 39-51, (2017)
[6] Dynkin, E. B., The space of exits of a Markov process, Uspekhi Mat. Nauk, 24, 89-152, (1969) · Zbl 0201.19801
[7] Margulis, G. A., Positive harmonic functions on nilpotent groups, Dokl. Akad. Nauk SSSR, 166, 1054-1057, (1966)
[8] Molchanov, S. A., On Martin boundaries for the direct product of Markov chains, Teor. Veroyatn. Primen., 12, 353-358, (1967) · Zbl 0292.60119
[9] Kai Lai Chung, Markov Chains with Stationary Transition Probabilities, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. · Zbl 0092.34304
[10] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word Processing in Groups, Jones and Bartlett Publishers, Boston, MA, 1992. · Zbl 0764.20017
[11] G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton Univ. Press, Princeton, NJ, 1989. · Zbl 0682.43001
[12] M. Gromov, Geometric Group Theory, Vol. 2: Asymptotic Invariants of Infinite Groups, London Math. Soc. Lecture Note Series, 182, Cambridge University Press, Cambridge, 1993.
[13] Guivarc’h, Y.; Raugi, A., Frontière de Furstenberg, propriétés de contraction et théorème de convergence, Z. Wahrsch. Verw. Gebiete, 69, 187-242, (1985) · Zbl 0558.60009
[14] Kaimanovich, V. A.; Vershik, A. M., Random walks on discrete groups: boundary and entropy, Ann. Probab., 11, 457-490, (1983) · Zbl 0641.60009
[15] Kaimanovich, V. A.; Woess, W., Boundary and entropy of space homogeneous Markov chains, Ann. Probab., 30, 323-363, (2002) · Zbl 1021.60056
[16] Vershik, A. M., Intrinsic metric on graded graphs, standardness, and invariant measures, Zap. Nauchn. Sem. POMI, 421, 58-67, (2014) · Zbl 1336.28014
[17] A. M. Vershik and A. V. Malyutin, “The absolute of finitely generated groups: I. Commutative (semi)groups,” to appear in Eur. J. Math. · Zbl 1403.28014
[18] W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts inMath., vol. 138, Cambridge Univ. Press, Cambridge, 2000. · Zbl 1142.60003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.