Unimodular measures on the space of all Riemannian manifolds

Abért, Miklós¹ ; Biringer, Ian²

¹ Alfréd Rényi Institute of Mathematics, Budapest, Hungary
² Department of Mathematics, Boston College, Chestnut Hill, MA, United States

Geometry & topology, Tome 26 (2022) no. 5, pp. 2295-2404.

Résumé

We study unimodular measures on the space ${\mathcal{ℳ}}^d$ of all pointed Riemannian $d$–manifolds. Examples can be constructed from finite-volume manifolds, from measured foliations with Riemannian leaves, and from invariant random subgroups of Lie groups. Unimodularity is preserved under weak${}^{\ast }$ limits, and under certain geometric constraints (eg bounded geometry) unimodular measures can be used to compactify sets of finite-volume manifolds. One can then understand the geometry of manifolds $M$ with large, finite volume by passing to unimodular limits.

We develop a structure theory for unimodular measures on ${\mathcal{ℳ}}^d$, characterizing them via invariance under a certain geodesic flow, and showing that they correspond to transverse measures on a foliated “desingularization” of ${\mathcal{ℳ}}^d$. We also give a geometric proof of a compactness theorem for unimodular measures on the space of pointed manifolds with pinched negative curvature, and characterize unimodular measures supported on hyperbolic $3$–manifolds with finitely generated fundamental group.

Détail

DOI : 10.2140/gt.2022.26.2295

Keywords: unimodular measures, Benjamini–Schramm convergence, invariant random subgroup, hyperbolic geometry, Lie group

Bibliographie

[1] M Abért, N Bergeron, I Biringer, T Gelander, Convergence of normalized Betti numbers in nonpositive curvature, preprint (2018)

[2] M Abért, N Bergeron, I Biringer, T Gelander, N Nikolov, J Raimbault, I Samet, On the growth of Betti numbers of locally symmetric spaces, C. R. Math. Acad. Sci. Paris 349 (2011) 831 | DOI

[3] M Abért, N Bergeron, I Biringer, T Gelander, N Nikolov, J Raimbault, I Samet, On the growth of L2–invariants for sequences of lattices in Lie groups, Ann. of Math. 185 (2017) 711 | DOI

[4] M Abért, N Bergeron, E Le Masson, Eigenfunctions and random waves in the Benjamini–Schramm limit, preprint (2018)

[5] M Abért, Y Glasner, B Virág, Kesten’s theorem for invariant random subgroups, Duke Math. J. 163 (2014) 465 | DOI

[6] M Abért, T Hubai, Benjamini–Schramm convergence and the distribution of chromatic roots for sparse graphs, Combinatorica 35 (2015) 127 | DOI

[7] I Agol, Tameness of hyperbolic 3–manifolds, preprint (2004)

[8] H Aimar, B Iaffei, L Nitti, On the Macías–Segovia metrization of quasi-metric spaces, Rev. Un. Mat. Argentina 41 (1998) 67

[9] D Aldous, R Lyons, Processes on unimodular random networks, Electron. J. Probab. 12 (2007) | DOI

[10] J A Álvarez López, R Barral Lijó, Bounded geometry and leaves, Math. Nachr. 290 (2017) 1448 | DOI

[11] J A Álvarez López, R Barral Lijó, A Candel, A universal Riemannian foliated space, Topology Appl. 198 (2016) 47 | DOI

[12] J A Álvarez López, A Candel, On turbulent relations, Fund. Math. 242 (2018) 207 | DOI

[13] W Ballmann, Lectures on Kähler manifolds, Eur. Math. Soc. (2006) | DOI

[14] W Ballmann, M Gromov, V Schroeder, Manifolds of nonpositive curvature, 61, Birkhäuser (1985) | DOI

[15] G Beer, Upper semicontinuous functions and the Stone approximation theorem, J. Approx. Theory 34 (1982) 1 | DOI

[16] R Benedetti, C Petronio, Lectures on hyperbolic geometry, Springer (1992) | DOI

[17] I Benjamini, N Curien, Ergodic theory on stationary random graphs, Electron. J. Probab. 17 (2012) | DOI

[18] I Benjamini, R Lyons, Y Peres, O Schramm, Group-invariant percolation on graphs, Geom. Funct. Anal. 9 (1999) 29 | DOI

[19] I Benjamini, R Lyons, O Schramm, Unimodular random trees, Ergodic Theory Dynam. Systems 35 (2015) 359 | DOI

[20] I Benjamini, O Schramm, Recurrence of distributional limits of finite planar graphs, Electron. J. Probab. 6 (2001) | DOI

[21] I Biringer, J Raimbault, Ends of unimodular random manifolds, Proc. Amer. Math. Soc. 145 (2017) 4021 | DOI

[22] I Biringer, J Souto, Ranks of mapping tori via the curve complex, J. Reine Angew. Math. 748 (2019) 153 | DOI

[23] I Biringer, O Tamuz, Unimodularity of invariant random subgroups, Trans. Amer. Math. Soc. 369 (2017) 4043 | DOI

[24] F Bonahon, Bouts des variétés hyperboliques de dimension 3, Ann. of Math. 124 (1986) 71 | DOI

[25] N Bourbaki, Integration, II : Chapters 7–9, Springer (2004) | DOI

[26] L Bowen, Cheeger constants and L2–Betti numbers, Duke Math. J. 164 (2015) 569 | DOI

[27] L Bowen, Invariant random subgroups of the free group, Groups Geom. Dyn. 9 (2015) 891 | DOI

[28] L Bowen, R Grigorchuk, R Kravchenko, Invariant random subgroups of lamplighter groups, Israel J. Math. 207 (2015) 763 | DOI

[29] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[30] S Brofferio, M Salvatori, W Woess, Brownian motion and harmonic functions on Sol(p,q), Int. Math. Res. Not. 2012 (2012) 5182 | DOI

[31] D Calegari, D Gabai, Shrinkwrapping and the taming of hyperbolic 3–manifolds, J. Amer. Math. Soc. 19 (2006) 385 | DOI

[32] R D Canary, Ends of hyperbolic 3–manifolds, J. Amer. Math. Soc. 6 (1993) 1 | DOI

[33] R D Canary, A covering theorem for hyperbolic 3–manifolds and its applications, Topology 35 (1996) 751 | DOI

[34] R D Canary, D B A Epstein, P Green, Notes on notes of Thurston, from: "Analytical and geometric aspects of hyperbolic space" (editor D B A Epstein), Lond. Math. Soc. Lect. Note Ser. 111, Cambridge Univ. Press (1987) 3

[35] A Candel, The harmonic measures of Lucy Garnett, Adv. Math. 176 (2003) 187 | DOI

[36] A Candel, L Conlon, Foliations, I, 23, Amer. Math. Soc. (2000) | DOI

[37] A Candel, L Conlon, Foliations, II, 60, Amer. Math. Soc. (2003) | DOI

[38] P J Catuogno, D S Ledesma, P R Ruffino, Foliated stochastic calculus: harmonic measures, Trans. Amer. Math. Soc. 368 (2016) 563 | DOI

[39] J Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970) 61 | DOI

[40] J Cheeger, Critical points of distance functions and applications to geometry, from: "Geometric topology: recent developments" (editors P De Bartolomeis, F Tricerri), Lecture Notes in Math. 1504, Springer (1991) 1 | DOI

[41] J Cheeger, D G Ebin, Comparison theorems in Riemannian geometry, 9, North-Holland (1975)

[42] A Connes, A survey of foliations and operator algebras, from: "Operator algebras and applications, I" (editor R V Kadison), Proc. Sympos. Pure Math. 38, Amer. Math. Soc. (1982) 521 | DOI

[43] D J Daley, D Vere-Jones, An introduction to the theory of point processes, I : Elementary theory and methods, Springer (2003) | DOI

[44] P Eberlein, Geodesic flows on negatively curved manifolds, II, Trans. Amer. Math. Soc. 178 (1973) 57 | DOI

[45] G A Edgar, A noncompact Choquet theorem, Proc. Amer. Math. Soc. 49 (1975) 354 | DOI

[46] A Eisenmann, Y Glasner, Generic IRS in free groups, after Bowen, Proc. Amer. Math. Soc. 144 (2016) 4231 | DOI

[47] J Feldman, C C Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, I, Trans. Amer. Math. Soc. 234 (1977) 289 | DOI

[48] R Feres, A Zeghib, Leafwise holomorphic functions, Proc. Amer. Math. Soc. 131 (2003) 1717 | DOI

[49] M Freedman, J Hass, P Scott, Least area incompressible surfaces in 3–manifolds, Invent. Math. 71 (1983) 609 | DOI

[50] D H Fremlin, Measure theory, IV: Topological measure spaces, part I, II, Torres Fremlin (2006)

[51] H Furstenberg, A note on Borel’s density theorem, Proc. Amer. Math. Soc. 55 (1976) 209 | DOI

[52] D Gaboriau, R Lyons, A measurable-group-theoretic solution to von Neumann’s problem, Invent. Math. 177 (2009) 533 | DOI

[53] L Garnett, Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal. 51 (1983) 285 | DOI

[54] L Garnett, Statistical properties of foliations, from: "Geometric dynamics" (editor J Palis Jr.), Lecture Notes in Math. 1007, Springer (1983) 294 | DOI

[55] É Ghys, Topologie des feuilles génériques, Ann. of Math. 141 (1995) 387 | DOI

[56] Y Glasner, Invariant random subgroups of linear groups, Israel J. Math. 219 (2017) 215 | DOI

[57] G Greschonig, K Schmidt, Ergodic decomposition of quasi-invariant probability measures, Colloq. Math. 84 (2000) 495 | DOI

[58] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) | DOI

[59] O Häggström, Infinite clusters in dependent automorphism invariant percolation on trees, Ann. Probab. 25 (1997) 1423 | DOI

[60] Y Hartman, O Tamuz, Furstenberg entropy realizations for virtually free groups and lamplighter groups, J. Anal. Math. 126 (2015) 227 | DOI

[61] Y Hartman, O Tamuz, Stabilizer rigidity in irreducible group actions, Israel J. Math. 216 (2016) 679 | DOI

[62] H Hatami, L Lovász, B Szegedy, Limits of locally-globally convergent graph sequences, Geom. Funct. Anal. 24 (2014) 269 | DOI

[63] E Heintze, H C Im Hof, Geometry of horospheres, J. Differential Geom. 12 (1977) 481

[64] S Helgason, Differential geometry, Lie groups, and symmetric spaces, 80, Academic (1978)

[65] V A Kaimanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funktsional. Anal. i Prilozhen. 22 (1988) 82

[66] V A Kaimanovich, Invariance, quasi-invariance and unimodularity for random graphs, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 441 (2015) 210

[67] V A Kaimanovich, M Lyubich, Conformal and harmonic measures on laminations associated with rational maps, 820, Amer. Math. Soc. (2005) | DOI

[68] M Kapovich, Hyperbolic manifolds and discrete groups, 183, Birkhäuser (2001) | DOI

[69] A Katok, V Climenhaga, Lectures on surfaces: (almost) everything you wanted to know about them, 46, Amer. Math. Soc. (2008) | DOI

[70] P Lessa, Brownian motion on stationary random manifolds, PhD thesis, Université Pierre et Marie Curie–Paris VI (2014)

[71] P Lessa, Reeb stability and the Gromov–Hausdorff limits of leaves in compact foliations, Asian J. Math. 19 (2015) 433 | DOI

[72] P Lessa, Brownian motion on stationary random manifolds, Stoch. Dyn. 16 (2016) | DOI

[73] L Lovász, Large networks and graph limits, 60, Amer. Math. Soc. (2012) | DOI

[74] W Lück, Approximating L2–invariants and homology growth, Geom. Funct. Anal. 23 (2013) 622 | DOI

[75] R Lyons, Y Peres, Probability on trees and networks, 42, Cambridge Univ. Press (2016) | DOI

[76] K Matsuzaki, M Taniguchi, Hyperbolic manifolds and Kleinian groups, Oxford Univ. Press (1998)

[77] C T Mcmullen, Renormalization and 3–manifolds which fiber over the circle, 142, Princeton Univ. Press (1996) | DOI

[78] K P Mok, On the differential geometry of frame bundles of Riemannian manifolds, J. Reine Angew. Math. 302 (1978) 16 | DOI

[79] C C Moore, C Schochet, Global analysis on foliated spaces, 9, Springer (1988) | DOI

[80] H Namazi, P Pankka, J Souto, Distributional limits of Riemannian manifolds and graphs with sublinear genus growth, Geom. Funct. Anal. 24 (2014) 322 | DOI

[81] P Petersen, Riemannian geometry, 171, Springer (2006) | DOI

[82] R R Phelps, Lectures on Choquet’s theorem, 1757, Springer (2001) | DOI

[83] J G Ratcliffe, Foundations of hyperbolic manifolds, 149, Springer (1994) | DOI

[84] L Rifford, A Morse–Sard theorem for the distance function on Riemannian manifolds, Manuscripta Math. 113 (2004) 251 | DOI

[85] V A Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949) 107

[86] S Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (1958) 338 | DOI

[87] M Schrödl-Baumann, l2–Betti numbers of random rooted simplicial complexes, Manuscripta Math. 162 (2020) 283 | DOI

[88] D Simmons, Conditional measures and conditional expectation ; Rohlin’s disintegration theorem, Discrete Contin. Dyn. Syst. 32 (2012) 2565 | DOI

[89] S Thomas, R Tucker-Drob, Invariant random subgroups of strictly diagonal limits of finite symmetric groups, Bull. Lond. Math. Soc. 46 (2014) 1007 | DOI

[90] W Thurston, Geometry and topology of three-manifolds, lecture notes (1980)

[91] W P Thurston, Hyperbolic structures on 3–manifolds, I : Deformation of acylindrical manifolds, Ann. of Math. 124 (1986) 203 | DOI

[92] V S Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963) 191 | DOI

[93] F Waldhausen, On irreducible 3–manifolds which are sufficiently large, Ann. of Math. 87 (1968) 56 | DOI

[94] F W Warner, Extensions of the Rauch comparison theorem to submanifolds, Trans. Amer. Math. Soc. 122 (1966) 341 | DOI