https://m2.mtmt.hu/ hun 2026-05-01 14:28 JournalArticle 33706737 APPROVED true Export Date: 20 March 2023 Correspondence Address: Abért, M.; Alfréd Rényi Institute of Mathematics, Realtanoda Utca 13-15, Hungary; email: abert@renyi.hu Funding details: NKFI-139502 Funding details: European Research Council, ERC, 648017 Funding text 1: This work was partially supported by ERC Consolidator Grant 648017 and the KKP NKFI-139502 Grant. Acknowledgements 0 false 2026-02-24T00:41:30.829+0000 2026-02-23T18:42:15.504+0000 2023-03-20T09:27:57.418+0000 true 10084481 Toroczkai Annamária /api/admin/10084481 Admin Toroczkai Annamária (idézők, admin) true 2026-02-23T18:42:15.772+0000 2026-02-23T18:42:15.772+0000 true false true 24 24 /api/publicationtype/24 PublicationType Folyóiratcikk 1 true 24 JournalArticle true 10000059 Article true 24 24 /api/publicationtype/24 PublicationType Folyóiratcikk 1 true 24 JournalArticle /api/subtype/10000059 Szakcikk SubType Szakcikk (Folyóiratcikk) 101 true 10000059 true 1 /api/category/1 Category Tudományos true 1 Abért, M. true true /api/journal/1417131 REVIEWED ISRAEL JOURNAL OF MATHEMATICS 0021-2172 1565-8511 true false 1417131 false 1417131 true 0021-2172 1565-8511 Journal FOREIGN 251 1 48 155 48 108 2022 Let G be a locally compact, second countable, unimodular group that is nondiscrete and noncompact. We explore the ergodic theory of invariant point processes on G. Our first result shows that every free probability measure preserving (pmp) action of G can be realized by an invariant point process. We then analyze the cost of pmp actions of G using this language. We show that among free pmp actions, the cost is maximal on the Poisson processes. This follows from showing that every free point process weakly factors onto any Poisson process and that the cost is monotone for weak factors, up to some restrictions. We apply this to show that G × ℤ has fixed price 1, solving a problem of Carderi. We also show that when G is a semisimple real Lie group, the rank gradient of any Farber sequence of lattices in G is dominated by the cost of the Poisson process of G. The same holds for the symmetric space X of G. This, in particular, implies that if the cost of the Poisson process of the hyperbolic 3-space ℍ3 vanishes, then the ratio of the Heegaard genus and the rank of a hyperbolic 3-manifold tends to infinity over arbitrary expander Farber sequences, in particular, the ratio can get arbitrarily large. On the other hand, if the cost of the Poisson process on ℍ3 does not vanish, it solves the cost versus L2 Betti problem of Gaboriau for countable equivalence relations. © 2022, The Hebrew University of Jerusalem. 2023 true true true false false NONE 2026-02-24 false 3 0 3 2 0 3 3 2 3 3 3 2 2 2 1 0 3 2 3 2 0 5 5 Folyóiratcikk Journal Article 24 3 Könyvrészlet Chapter in Book 25 0 Könyv Book 23 0 Egyéb konferenciaközlemény Conference paper 31 0 Egyéb konferenciakötet Conference proceedings 32 0 Oltalmi formák Protection forms 26 0 Disszertáció Thesis 28 0 Egyéb Miscellaneous 29 0 Alkotás Achievement 22 0 Kutatási adat Research data 33 0 Folyóiratcikk Journal Article 24 0 0 0 Könyvrészlet Chapter in Book 25 0 0 0 Könyv Book 23 0 0 0 Egyéb konferenciaközlemény Conference paper 31 0 0 0 Egyéb konferenciakötet Conference proceedings 32 0 0 0 Oltalmi formák Protection forms 26 0 0 0 Disszertáció Thesis 28 0 0 0 Egyéb Miscellaneous 29 0 0 0 Alkotás Achievement 22 0 0 0 Kutatási adat Research data 33 0 0 0 2023 0 1 1 1 1 2024 0 0 0 0 0 2025 0 2 1 2 1 Q1 false 10011747 false false false 2023-06-18T08:29:54.911+0000 Algebra (HRN RAMKI) 2 8 1 2 0 true Language 10002 /api/language/10002 Angol Angol English true 2 true PersonAuthorship 107065184 /api/authorship/107065184 Abért, M. ✉ [Abért, Miklós (Csoportelmélet), szerző] Algebra (HRN RAMKI) 1 0.5 true false true Author 10011747 /api/author/10011747 Abért Miklós (Csoportelmélet) Abért Miklós true 10011747 true Abért M. true false false AuthorshipType 1 /api/authorshiptype/1 Szerző 0 true 0 true false true PersonAuthorship 107065185 /api/authorship/107065185 Mellick, S. [Mellick, Samuel (Gráfelmélet), szerző] 2 0.5 false true false Author 10066346 /api/author/10066346 Mellick Samuel (Gráfelmélet) Mellick Samuel true true Mellick S. true false false AuthorshipType 1 /api/authorshiptype/1 Szerző 0 true 0 true false true PublicationIdentifier 23254151 /api/publicationidentifier/23254151 DOI: 10.1007/s11856-022-2445-9 PlainSource 6 /api/publicationsource/6 DOI PublicationSourceType 10001 /api/publicationsourcetype/10001 DOI true true true DOI DOI https://doi.org/@@@ true true 6 true IDENTICAL 10.1007/s11856-022-2445-9 https://doi.org/10.1007/s11856-022-2445-9 false true PublicationIdentifier 23254152 /api/publicationidentifier/23254152 WoS: 000943492500002 PlainSource 1 /api/publicationsource/1 WoS PublicationSourceType 10003 /api/publicationsourcetype/10003 Indexelő adatbázis false true true WoS WoS https://www.webofscience.com/wos/woscc/full-record/@@@ true true 1 true IDENTICAL 000943492500002 https://www.webofscience.com/wos/woscc/full-record/000943492500002 false true PublicationIdentifier 23254150 /api/publicationidentifier/23254150 Scopus: 85149592369 PlainSource 3 /api/publicationsource/3 Scopus PublicationSourceType 10003 /api/publicationsourcetype/10003 Indexelő adatbázis false true true Scopus Scopus http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-@@@ true true 3 true IDENTICAL 85149592369 http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85149592369 false true SjrRating 11306502 /api/sjrrating/11306502 sjr:Q1 (2022) Scopus - Mathematics (miscellaneous) ISRAEL JOURNAL OF MATHEMATICS 0021-2172 1565-8511 73 0.24 journal RatingType 10002 /api/ratingtype/10002 sjr sjr true true ClassificationExternal 2601 /api/classificationexternal/2601 Scopus - Mathematics (miscellaneous) true 2601 true Q1 DIRECT true true Reference 39413352 /api/reference/39413352 1. Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J., Samet, I., On the growth of L2-invariants for sequences of lattices in Lie groups (2017) Annals of Mathematics, 185, pp. 711-790 1 false true Reference 39413353 /api/reference/39413353 2. Anantharaman-Delaroche, C., Invariant proper metrics on coset spaces (2013) Topology and its Applications, 160, pp. 546-552 2 false true Reference 39413354 /api/reference/39413354 3. Abert, M., Gelander, T., Nikolov, N., Rank, combinatorial cost, and homology torsion growth in higher rank lattices (2017) Duke Mathematical Journal, 166, pp. 2925-2964 3 false true Reference 39413355 /api/reference/39413355 4. Abért, M., Nikolov, N., Rank gradient, cost of groups and the rank versus Heegaard genus problem (2012) Journal of the European Mathematical Society, 14, pp. 1657-1677 4 false true Reference 39413356 /api/reference/39413356 5. Avni, N., Spectral and mixing properties of actions of amenable groups (2005) Electronic Research Announcements of the American Mathematical Society, 11, pp. 57-63 5 false true Reference 39413357 /api/reference/39413357 6. Abért, M., Weiss, B., Bernoulli actions are weakly contained in any free action (2013) Ergodic theory and dynamical systems, 33, pp. 323-333 6 false true Reference 39413358 /api/reference/39413358 7. Babson, E., Benjamini, I., Cut sets and normed cohomology with applications to percolation (1999) Proceedings of the American Mathematical Society, 127, pp. 589-597 7 false true Reference 39413359 /api/reference/39413359 8. Bader, U., Caprace, P.-E., Gelander, T., Mozes, S., Simple groups without lattices (2012) Bulletin of the London Mathematical Society, 44, pp. 55-67 8 false true Reference 39413360 /api/reference/39413360 9. Bowen, L., Hoff, D., Ioana, A., von Neumann’s problem and extensions of nonamenable equivalence relations, Groups (2018) Geometry, and Dynamics, 12, pp. 399-448 9 false true Reference 39413361 /api/reference/39413361 10. Burton, P., Kechris, A., Weak containment of measure-preserving group actions (2020) Ergodic Theory and Dynamical Systems, 40, pp. 2681-2733 10 false true Reference 39413362 /api/reference/39413362 11. Blaszczyszyn, B., (2017) Lecture Notes on Random Geometric Models—Random Graphs, , https://hal.inria.fr/cel-01654766/document, Point Processes and Stochastic Geometry, Lecture, December 11 false true Reference 39413363 /api/reference/39413363 12. Bowen, L., All properly ergodic markov chains over a free group are orbit equivalent (2018) Unimodularity in Randomly Generated Graphs, 719, pp. 155-174. , Contemporary Mathematics, Ameerican Mathematical Society, Providence, RI 12 false true Reference 39413364 /api/reference/39413364 13. Carderi, A., Asymptotic invariants of lattices in locally compact groups, , https://arxiv.org/abs/1812.02133 13 false true Reference 39413365 /api/reference/39413365 14. Cornulier, Y., de la Harpe, P., (2016) Metric Geometry of Locally Compact Groups, 25. , EMS Tracts Mathematics, European Mathematical Society, Zürich 14 false true Reference 39413366 /api/reference/39413366 15. Conley, C.T., Gaboriau, D., Marks, A.S., Tucker-Drob, R.D., One-ended spanning subforests and treeability of groups, , https://arxiv.org/abs/2104.07431 15 false true Reference 39413367 /api/reference/39413367 16. Coupier, D., Tran, V.C., The 2d-directed spanning forest is almost surely a tree (2013) Random Structures & Algorithms, 42, pp. 59-72 16 false true Reference 39413368 /api/reference/39413368 17. Daley, D.J., Vere-Jones, D., (2003) An Introduction to the Theory of Point Processes. Vol. I, , Probability and its Applications (New York), Springer, New York 17 false true Reference 39413369 /api/reference/39413369 18. Daley, D.J., Vere-Jones, D., (2008) An Introduction to the Theory of Point Processes. Vol. II, , Probability and its Applications (New York), Springer, New York 18 false true Reference 39413370 /api/reference/39413370 19. Forrest, P., On the virtual groups defined by ergodic actions of R n and Z n (1974) Advances in Mathematics, 14, pp. 271-308 19 false true Reference 39413371 /api/reference/39413371 20. Fraczyk, M., Growth of mod −2 homology in higher-rank locally symmetric spaces (2022) Duke Mathematical Journal, 171, pp. 247-271 20 false true Reference 39413372 /api/reference/39413372 21. Furman, A., A survey of measured group theory (2011) Geometry, Rigidity, and Group Actions, pp. 296-374. , Chicago Lectures Mathematics, University of Chicago Press, Chicago, IL 21 false true Reference 39413373 /api/reference/39413373 22. Gaboriau, D., Around the orbit equivalence theory of the free groups, cost and ℓ2 Betti numbers, , https://perso.ens-lyon.fr/gaboriau/Travaux-Publi/ME-Cost-L2-Lectures/ME-Cost-L2-Lectures.html 22 false true Reference 39413374 /api/reference/39413374 23. Gaboriau, D., Coût des relations d’équivalence et des groupes (2000) Inventiones Mathematicae, 139, pp. 41-98 23 false true Reference 39413375 /api/reference/39413375 24. Gaboriau, D., Invariants ℓ2 de relations d’équivalence et de groupes (2002) Publications Mathématiques. Institut de Hautes Études Scientifiques, 95, pp. 93-150 24 false true Reference 39413376 /api/reference/39413376 25. Gaboriau, D., What is cost? (2010) Notices of the American Mathematical Society, 57, pp. 1295-1296 25 false true Reference 39413377 /api/reference/39413377 26. Gelander, T., Volume versus rank of lattices (2011) Journal für die Reine und Angewandte Mathematik, 661, pp. 237-248 26 false true Reference 39413378 /api/reference/39413378 27. Greschonig, S.K., Gernot, Ergodic decomposition of quasi-invariant probability measures (2000) Colloquium Mathematicum, 84-85, pp. 495-514 27 false true Reference 39413379 /api/reference/39413379 28. Gurel-Gurevich, O., Peled, R., Poisson thickening (2013) Israel Journal of Mathematics, 196, pp. 215-234 28 false true Reference 39413380 /api/reference/39413380 29. Gentner, D., Last, G., Palm pairs and the general mass-transport principle (2011) Mathematische Zeitschrift, 267, pp. 695-716 29 false true Reference 39413381 /api/reference/39413381 30. Holroyd, A.E., Lyons, R., Soo, T., Poisson splitting by factors (2011) Annals of Probability, 39, pp. 1938-1982 30 false true Reference 39413382 /api/reference/39413382 31. Holroyd, A., Peres, Y., Trees and matchings from point processes (2003) Electronic Communications in Probability, 8, pp. 17-27 31 false true Reference 39413383 /api/reference/39413383 32. Holroyd, A.E., Peres, Y., Extra heads and invariant allocations (2005) Annals of Probability, 33, pp. 31-52 32 false true Reference 39413384 /api/reference/39413384 33. Kallenberg, O., Invariant palm and related disintegrations via skew factorization (2011) Probability Theory and Related Fields, 149, pp. 279-301 33 false true Reference 39413385 /api/reference/39413385 34. Kallenberg, O., (2017) Random Measures, 77. , Theory and Applications, Probability Theory and Stochastic Modelling, Springer, Cham 34 false true Reference 39413386 /api/reference/39413386 35. Kechris, A.S., The Theory of Countable Borel Equivalence Relations, , http://math.caltech.edu/ 35 false true Reference 39413387 /api/reference/39413387 36. Kingman, J.F.C., (1993) Poisson Processes, 3. , Oxford Studies Probability, The Clarendon Press, Oxford University Press, New York 36 false true Reference 39413388 /api/reference/39413388 37. Kechris, A.S., Miller, B.D., (2004) Topics in Orbit Equivalence, 1852. , Lecture Notes Mathematics, Springer, Berlin 37 false true Reference 39413389 /api/reference/39413389 38. Kyed, D., Petersen, H.D., Vaes, S., L 2 -Betti numbers of locally compact groups and their cross section equivalence relations (2015) Transactions of the American Mathematical Society, 367, pp. 4917-4956 38 false true Reference 39413390 /api/reference/39413390 39. Kechris, A., Solecki, S., Todorcevic, S., Borel chromatic numbers (1999) Advances in Mathematics, 141, pp. 1-44 39 false true Reference 39413391 /api/reference/39413391 40. Lackenby, M., Heegaard splittings, the virtually Haken conjecture and property (2006) Inventiones Mathematicae, 164, pp. 317-359 40 false true Reference 39413392 /api/reference/39413392 41. Li, T., Rank and genus of 3-manifolds (2013) Journal of the American Mathematical Society, 26, pp. 777-829 41 false true Reference 39413393 /api/reference/39413393 42. Last, G., Penrose, M.D., Poisson process Fock space representation, chaos expansion and covariance inequalities (2011) Probability Theory and Related Fields, 150, pp. 663-690 42 false true Reference 39413394 /api/reference/39413394 43. Lyons, R., Peres, Y., (2016) Probability on Trees and Networks, 42. , Cambridge Series Statistical and Probabilistic Mathematics, Cambridge University Press, New York 43 false true Reference 39413395 /api/reference/39413395 44. Last, G., Penrose, M., (2018) Lectures on the Poisson Process, 7. , Institute of Mathematical Statistics Textbooks, Cambridge University Press, Cambridge 44 false true Reference 39413396 /api/reference/39413396 45. Lubotzky, A., Slutsky, R., On the asymptotic number of generators of high rank arithmetic lattices (2022) Michigan Mathematical Journal, 72, pp. 465-477 45 false true Reference 39413397 /api/reference/39413397 46. Mellick, S., The Palm Groupoid of a Point Process and Factor Graphs on Amenable and Property (T) Groups, , https://arxiv.org/abs/2101.07238 46 false true Reference 39413398 /api/reference/39413398 47. Meester, R., Roy, R., (1996) Continuum Percolation, 119. , Cambridge Tracts Mathematics, Cambridge University Press, Cambridge 47 false true Reference 39413399 /api/reference/39413399 48. Ornstein, D.S., Weiss, B., Entropy and isomorphism theorems for actions of amenable groups (1987) Journal d’Analyse Mathématique, 48, pp. 1-141 48 false true Reference 39413400 /api/reference/39413400 49. Slutsky, K., Lebesgue orbit equivalence of multidimensional Borel flows: A picturebook of tilings (2017) Ergodic Theory and Dynamical Systems, 37, pp. 1966-1996 49 false true Reference 39413401 /api/reference/39413401 50. Soo, T., Wilkens, A., Finitary isomorphisms of poisson point processes (2019) Annals of Probability, 47, pp. 3055-3081 50 false true Reference 39413402 /api/reference/39413402 51. Timar, A., Tree and grid factors of general point processes (2004) Electronic Communications in Probability, 9, pp. 53-59 51 false true Reference 39413403 /api/reference/39413403 52. Zimmer, R.J., (1984) Ergodic Theory and Semisimple Groups, 81. , Monographs Mathematics, Birkhäuser, Basel 52 false true /api/publication/33706737 Abért M. et al. Point processes, cost, and the growth of rank in locally compact groups. (2022) ISRAEL JOURNAL OF MATHEMATICS 0021-2172 1565-8511 251 1 48-155<div class="JournalArticle Publication long-list"> <div class="authors"> <img title="Forrásközlemény" style="float: left" src="/frontend/resources/grid/publication-core-icon.png"> <img title="Idézőközlemény" style="float: left" src="/frontend/resources/grid/publication-citation-icon.png"> <div class="autype autype0"> <span class="author-name" mtid="10011747"><a href="/gui2/?type=authors&mode=browse&sel=10011747" target="_blank">Abért M. ✉ (<span class="authorship-author-name">Abért Miklós</span> <span class="authorAux-mtmt"> Csoportelmélet</span>) </a> </span> <span class="author-affil"><span title="Rényi Alfréd Matematikai Kutatóintézet">RAMKI</span>/Algebra</span> ;&nbsp;&nbsp;&nbsp; <span class="author-name" mtid="10066346"><a href="/gui2/?type=authors&mode=browse&sel=10066346" target="_blank">Mellick S. (<span class="authorship-author-name">Mellick Samuel</span> <span class="authorAux-mtmt"> Gráfelmélet</span>) </a> </span> </div> </div> <div class="title"><a href="/gui2/?mode=browse&params=publication;33706737" target="_blank">Point processes, cost, and the growth of rank in locally compact groups</a></div> <div> <span class="journal-title">ISRAEL JOURNAL OF MATHEMATICS</span> <span class="journal-issn">(<a target="_blank" href="https://portal.issn.org/resource/ISSN/0021-2172">0021-2172</a> <a target="_blank" href="https://portal.issn.org/resource/ISSN/1565-8511">1565-8511</a>)</span>: <span class="journal-volume">251</span> <span class="journal-issue">1</span> <span class="page"> pp 48-155 </span> <span class="year">(2022)</span> </div> <div class="pub-footer"> <span class="language" xmlns="http://www.w3.org/1999/html">Nyelv: Angol | </span> <span class="identifiers"> <span class="id identifier oa_none" title="none"> <a style="color:blue" title="10.1007/s11856-022-2445-9" target="_blank" href="https://doi.org/10.1007/s11856-022-2445-9"> DOI </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:blue" title="000943492500002" target="_blank" href="https://www.webofscience.com/wos/woscc/full-record/000943492500002"> WoS </a> </span> <span class="id identifier oa_none" title="none"> <a style="color:blue" title="85149592369" target="_blank" href="http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85149592369"> Scopus </a> </span> </span> <OnlyViewableByAuthor><div class="ratings"> <div class="journal-subject">Folyóirat szakterülete: Scopus - Mathematics (miscellaneous)&nbsp;&nbsp;&nbsp;SJR indikátor:&nbsp;Q1</div> </div></OnlyViewableByAuthor> <div class="publication-citation" style="margin-left: 0.5cm;"> <span title="Nyilvános idézőközlemények összesen, említések nélkül" class="citingPub-count">Nyilvános idéző összesen: 3</span> | Független: 2 | Függő: 1 | Nem jelölt: 0 | WoS jelölt: 2 | Scopus jelölt:&nbsp;3 | WoS/Scopus jelölt:&nbsp;3 | DOI jelölt:&nbsp;3 </div> <div class="publication-citation"> <a target="_blank" href="/api/publication?cond=citations.related;eq;33706737&sort=publishedYear,desc&sort=title"> Idézett közlemények száma: 5 </a> </div> <div class="mtid"><span class="long-pub-mtid">Közlemény: 33706737</span> | <span class="status-data status-APPROVED"> Nyilvános </span> Forrás Idéző | <span class="type-subtype">Folyóiratcikk ( Szakcikk ) </span> | <span class="pub-category">Tudományos</span> | <span class="publication-sourceOfData">Scopus</span> </div> <div class="lastModified">Utolsó módosítás: 2026.02.23. 19:42 Szakonyi Erzsebet (RAMKI admin 4) </div> <pre class="comment" style="margin-top: 0; margin-bottom: 0;"><u>Megjegyzés</u>: Export Date: 20 March 2023 Correspondence Address: Abért, M.; Alfréd Rényi Institute of Mathematics, Realtanoda Utca 13-15, Hungary; email: abert@renyi.hu Funding details: NKFI-139502 Funding details: European Research Council, ERC, 648017 Funding text 1: This work was partially supported by ERC Cons...</pre> </div></div>