×

Unimodularity of invariant random subgroups. (English) Zbl 1365.28014

Authors’ abstract: An invariant random subgroup \({H\leq G}\) is a random closed subgroup whose law is invariant to conjugation by all elements of \(G\) (they were first introduced by M. Abert et al. [Duke Math. J. 163, No. 3, 465–488 (2014; Zbl 1344.20061)] and A. M. Vershik [Mosc. Math. J. 12, No. 1, 193–212 (2012; Zbl 1294.37004)]). When \(G\) is locally compact and second countable, we show that for every invariant random subgroup \({H\leq G}\) there almost surely exists an invariant measure on \(G/H\). Equivalently, the modular function of \(H\) is almost surely equal to the modular function of \(G\), restricted to \(H\). We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups.

MSC:

28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ab{\'e}rt, Mikl{\'o}s; Bergeron, Nicolas; Biringer, Ian; Gelander, Tsachik; Nikolov, Nikolay; Raimbault, Jean; Samet, Iddo, On the growth of \({L}^2\)-invariants for sequences of lattices in lie groups, preprint, arXiv:1210.2961 (2012) · Zbl 1379.22006
[2] Ab{\'e}rt, Mikl{\'o}s; Biringer, Ian, Invariant random hyperbolic 3-manifolds with finitely generated \(\pi_1\) are doubly degenerate (2014)
[3] Ab{\'e}rt, Mikl{\'o}s; Glasner, Yair; Vir{\'a}g, B{\'a}lint, Kesten’s theorem for invariant random subgroups, Duke Math. J., 163, 3, 465-488 (2014) · Zbl 1344.20061 · doi:10.1215/00127094-2410064
[4] Aldous, David; Lyons, Russell, Processes on unimodular random networks, Electron. J. Probab., 12, no. 54, 1454-1508 (2007) · Zbl 1131.60003 · doi:10.1214/EJP.v12-463
[5] Bader, Uri; Duchesne, Bruno; L\'ecureux, Jean, Amenable invariant random subgroups, preprint, arXiv:1409.4745 (2014) · Zbl 1356.22006
[6] Benjamini, I.; Lyons, R.; Peres, Y.; Schramm, O., Group-invariant percolation on graphs, Geom. Funct. Anal., 9, 1, 29-66 (1999) · Zbl 0924.43002 · doi:10.1007/s000390050080
[7] Benjamini, Itai; Schramm, Oded, Percolation in the hyperbolic plane, J. Amer. Math. Soc., 14, 2, 487-507 (electronic) (2001) · Zbl 1037.82018 · doi:10.1090/S0894-0347-00-00362-3
[8] Benjamini, Itai; Schramm, Oded, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab., 6, no. 23, 13 pp. (electronic) pp. (2001) · Zbl 1010.82021 · doi:10.1214/EJP.v6-96
[9] Bergeron, N.; Gaboriau, D., Asymptotique des nombres de Betti, invariants \(l^2\) et laminations, Comment. Math. Helv., 79, 2, 362-395 (2004) · Zbl 1061.55005 · doi:10.1007/s00014-003-0798-1
[10] Bourbaki, Nicolas, Integration. {II}. {C}hapters 7-9, Elements of Mathematics (Berlin) (2004), Springer-Verlag: Berlin:Springer-Verlag · Zbl 1095.28002
[11] Bowen, Lewis, Random walks on random coset spaces with applications to Furstenberg entropy, Invent. Math., 196, 2, 485-510 (2014) · Zbl 1418.37003 · doi:10.1007/s00222-013-0473-0
[12] Bowen, Lewis, Invariant random subgroups of the free group, Groups Geom. Dyn., 9, 3, 891-916 (2015) · Zbl 1358.37011
[13] Bowen, Lewis; Grigorchuk, Rostislav; Kravchenko, Rostyslav, Invariant random subgroups of lamplighter groups, Israel J. Math., 207, 2, 763-782 (2015) · Zbl 1334.43006 · doi:10.1007/s11856-015-1160-1
[14] Creutz, Darren; Peterson, Jesse, Stabilizers of ergodic actions of lattices and commensurators, preprint, arXiv:1303.3949 (2013) · Zbl 1375.37007
[15] Feldman, Jacob; Moore, Calvin C., Ergodic equivalence relations, cohomology, and von Neumann algebras, Bull. Amer. Math. Soc., 81, 5, 921-924 (1975) · Zbl 0317.22002
[16] H{\`“a}ggstr{\'”o}m, Olle, Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab., 25, 3, 1423-1436 (1997) · Zbl 0895.60098 · doi:10.1214/aop/1024404518
[17] Hartman, Yair; Tamuz, Omer, Furstenberg entropy realizations for virtually free groups and lamplighter groups, J. Anal. Math., 126, 227-257 (2015) · Zbl 1358.37021 · doi:10.1007/s11854-015-0016-2
[18] Hartman, Yair; Tamuz, Omer, Stabilizer rigidity in irreducible group actions, preprint, arXiv:1307.7539 (2013) · Zbl 1356.22009
[19] Moore, Calvin C.; Schochet, Claude L., Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications 9, xiv+293 pp. (2006), Cambridge University Press, New York · Zbl 1091.58015
[20] Nachbin, Leopoldo, The Haar integral, xii+156 pp. (1965), D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London · Zbl 0131.37903
[21] Renault, Jean, A groupoid approach to \(C^{\ast } \)-algebras, Lecture Notes in Mathematics 793, ii+160 pp. (1980), Springer, Berlin · Zbl 0433.46049
[22] Simmons, David, Conditional measures and conditional expectation; Rohlin’s disintegration theorem, Discrete Contin. Dyn. Syst., 32, 7, 2565-2582 (2012) · Zbl 1244.28002 · doi:10.3934/dcds.2012.32.2565
[23] Stuck, Garrett; Zimmer, Robert J., Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2), 139, 3, 723-747 (1994) · Zbl 0836.22018 · doi:10.2307/2118577
[24] Varadarajan, V. S., Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc., 109, 191-220 (1963) · Zbl 0192.14203
[25] Vershik, A. M., Nonfree actions of countable groups and their characters, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). J. Math. Sci. (N. Y.), 378 174, 1, 1-6 (2011) · Zbl 1279.37004 · doi:10.1007/s10958-011-0273-2
[26] Vershik, A. M., Totally nonfree actions and the infinite symmetric group, Mosc. Math. J., 12, 1, 193-212, 216 (2012) · Zbl 1294.37004
[27] Zimmer, Robert J., Ergodic theory and semisimple groups, Monographs in Mathematics 81, x+209 pp. (1984), Birkh\"auser Verlag, Basel · Zbl 0571.58015 · doi:10.1007/978-1-4684-9488-4
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.