Skip to main content Accessibility help
×
Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-26T02:03:57.856Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  07 November 2024

Alexander S. Kechris
Affiliation:
California Institute of Technology
Get access
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2024

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ackerman, N., Freer, C. and Patel, R.. Invariant measures concentrated on countable structures. Forum Math. Sigma, 4 (2016), e17, 59pp. 129CrossRefGoogle Scholar
Adams, S.. Indecomposability of treed equivalence relations. Israel J. Math., 64(3) (1988), 362380. 50, 93, 99CrossRefGoogle Scholar
Adams, S.. An equivalence relation that is not freely generated. Proc. Amer. Math. Soc., 102 (1988), 565566. 95CrossRefGoogle Scholar
Adams, S.. Trees and amenable equivalence relations. Ergodic Theory Dynam. Systems, 10 (1990), 114. 93, 97CrossRefGoogle Scholar
Adams, S.. Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups. Topology, 33(4) (1994), 765783. 73CrossRefGoogle Scholar
Adams, S.. Indecomposability of equivalence relations generated by word hyperbolic groups. Topology, 33(4) (1994), 785798. 99CrossRefGoogle Scholar
Adams, S.. Containment does not imply Borel reducibility. In Set Theory, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 58, American Mathematical Society, 2002, 123. 17Google Scholar
Adams, S. and Kechris, A.S.. Linear algebraic groups and countable Borel equivalence relations. J. Amer. Math. Soc., 13(4) (2000), 909943. 51, 52, 55, 58CrossRefGoogle Scholar
Adams, S. and Lyons, R.. Amenability, Kazhdan’s property and percolation for trees, groups, and equivalence relations. Israel J. Math., 75 (1991), 341370. 87, 90CrossRefGoogle Scholar
Adams, S. and Spatzier, R.. Kazhdan groups, cocycles and trees. Amer. J. Math., 112 (1990), 271287. 100CrossRefGoogle Scholar
Allison, S.. Countable Borel treeable equivalence relations are classified by ℓ1. ArXiv: 2305.01049. 70Google Scholar
Ambrose, W.. Representation of ergodic flows. Ann. of Math., 42 (1941), 723739. 25CrossRefGoogle Scholar
Anantharaman, C. and Popa, S.. An Introduction to II1 Factors, math.ucla.edu/~popa/books.html. xiiGoogle Scholar
Andretta, A., Camerlo, R. and Hjorth, G.. Conjugacy equivalence relation on subgroups. Fund. Math., 167 (2001), 189212. 112, 113, 118CrossRefGoogle Scholar
Barwise, J.. Admissible Sets and Structures, Springer-Verlag, 1975. 31CrossRefGoogle Scholar
Becker, H. and Kechris, A.S.. The Descriptive Set Theory of Polish Group Actions, Cambridge University Press, 1996. 7, 12, 30, 31, 37CrossRefGoogle Scholar
Bernshteyn, A. and Yu, J.. Coarse embeddings into grids and asymptotic dimension for Borel graphs of polynomial growth. ArXiv:2302.04737. 70Google Scholar
Bezuglyi, S.I. and Golodets, V.Ya.. Hyperfinite and II1 actions for nonamenable groups. J. Funct. Anal., 40 (1981), 3044. 91CrossRefGoogle Scholar
Bowen, L.. Finitary random interlacements and the Gaboriau–Lyons problem. Geom. Funct. Anal., 29(3) (2019), 659689. 79CrossRefGoogle Scholar
Bowen, L., Hoff, D. and Ioana, A.. von Neumann’s problem and extensions of non-amenable equivalence relations. Groups Geom. Dyn., 12(2) (2018), 399448. 79CrossRefGoogle Scholar
Boykin, C.M. and Jackson, S.. Some Applications of Regular Markers. In Lecture Notes in Logic, 24, Assoc. Symb. Logic, 2006, 138155. 71Google Scholar
Boykin, C.M. and Jackson, S.. Borel boundedness and the lattice rounding property. Contemp. Math., 425 (2007), 113126. 66, 67, 68CrossRefGoogle Scholar
Brown, N.P. and Ozawa, N.. C-Algebras and Finite-Dimensional Approximations, Amer. Math. Soc., 2008. 136CrossRefGoogle Scholar
Calderoni, P.. Rotation equivalence and cocycle superrigidity for compact actions. J. London Math. Soc., 107(1) (2023), 189212. 56, 101CrossRefGoogle Scholar
Calderoni, F. and Clay, A.. Borel structures on the space of left-orderings. Bull. London Math. Soc., 54(1) (2022), 8394. 114CrossRefGoogle Scholar
Camerlo, R.. The relation of recursive isomorphism for countable structures. J. Symb. Logic, 167(2) (2002), 879875. 112CrossRefGoogle Scholar
Carrière, Y. and Ghys, E.. Relations d’équivalence moyennables sur les groupes de Lie. C.R. Acad. Sci. Paris Sér. I Math., 300(19) (1985), 677680. 66, 91Google Scholar
Chan, W. and Meehan, C.. Definable combinatorics of some Borel equivalence relations. ArXiv:1709.04567. 77Google Scholar
Chaube, P. and Nadkarni, M.G.. A version of Dye’s Theorem for descriptive dynamical systems. Sankyā, Ser. A, 49 (1987), 288304. 12Google Scholar
Chaube, P. and Nadkarni, M.G.. On orbit equivalence of Borel automorphisms. Proc. Indian Acad. Sci. Math. Sci., 99(3) (1989), 255261. 12CrossRefGoogle Scholar
Chen, R.. Borel structurability by locally finite simplicial complexes. Proc. Amer. Math. Soc., 146(7) (2018), 30853096. 105, 106CrossRefGoogle Scholar
Chen, R.. Decompositions and measures on countable Borel equivalence relations. Ergodic Theory Dynam. Systems, 41(12) (2021), 36713703. 39, 106, 125CrossRefGoogle Scholar
Chen, R.. On uniform ergodic decomposition. rynchn.github.io/math/. 39Google Scholar
Chen, R. and Kechris, A.S.. Structurable equivalence relations. Fund. Math., 242 (2018), 109185. See also ArXiv:1606.01995. 14, 85, 88, 95, 106, 126, 128, 129, 130CrossRefGoogle Scholar
Chen, R., Poulin, A., Tao, R. and Tserunyan, A.. Tree-like graphings, wallings, and median graphings of equivalence relations. ArXiv: 2308.13010. 99Google Scholar
Chen, R., Terlov, G. and Tserunyan, A.. Nonamenable subforests of multi-ended quasi-pmp graphs. ArXiv: 2211.07908. 97Google Scholar
Christensen, J.P.R.. Topology and Borel Structure, North-Holland, 1974. 91Google Scholar
Clemens, J.D.. Generating equivalence relations by homeomorphisms. https://citeseerx.ist.psu.edu/pdf/53c95108384476774591de17efc860df49a654bf, 2008. 6Google Scholar
Clemens, J.D.. Isomorphism of subshifts is a universal countable Borel equivalence relation. Israel J. Math., 170 (2009), 113123. 5, 73, 114CrossRefGoogle Scholar
Clemens, J.D.. Isomorphism and weak conjugacy of free Bernoulli subflows. Contemp. Math., 2020, 77–87. 114Google Scholar
Clemens, J.D., Conley, C. and Miller, B.D.. Borel homomorphisms of smooth σ-ideals. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=2255f1001874e0ab9943a964ee6c3a3fecfc2b15, 2007. 128Google Scholar
Clemens, J.D., Lecomte, D. and Miller, B.D.. Essential countability of treeable equivalence relations. Adv. Math., 265 (2014), 131. 29CrossRefGoogle Scholar
Conley, C.T., Gaboriau, D., Marks, A.S. and Tucker-Drob, R.D.. One-ended spanning subforests and treeability of groups. ArXiv:2104.07431. 90, 98, 99Google Scholar
Conley, C.T., Jackson, S., Marks, A.S., Seward, B. and Tucker-Drob, R.D.. Hyperfiniteness and Borel combinatorics. J. Eur. Math. Soc., 22(3) (2020), 877892. 77CrossRefGoogle Scholar
Conley, C.T., Jackson, S., Marks, A.S., Seward, B. and Tucker-Drob, R.D.. Borel asymptotic dimension and hyperfinite equivalence relations. Duke Math. J., to appear, ArXiv:2009.06721. 71Google Scholar
Conley, C.T., Kechris, A.S. and Miller, B.D.. Stationary probability measures and topological realizations. Israel J. Math., 198 (2013), 333345. 14, 37CrossRefGoogle Scholar
Conley, C.T. and Marks, A.S.. Distance from marker sequences in locally finite Borel graphs. Contemp. Math., 752 (2020), 8992. 11CrossRefGoogle Scholar
Conley, C.T. and Miller, B.D.. Measure reducibility of countable Borel equivalence relations. Ann. of Math., 185(2) (2017), 347402. 9, 16, 17, 51, 54, 64, 73, 78, 103, 104CrossRefGoogle Scholar
Conley, C.T. and Miller, B.D.. Incomparable actions of free groups. Ergodic Theory Dynam. Systems, 37 (2017), 20842098. 54, 103, 105CrossRefGoogle Scholar
Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergodic Theory Dynam. Systems, 1 (1981), 431450. 72, 89, 90, 91CrossRefGoogle Scholar
Coskey, S.. Descriptive aspects of torsion-free abelian groups. Ph.D. Thesis, Rutgers University, 2008. scoskey.org/publications. 58Google Scholar
Coskey, S.. Borel reductions of profinite actions of SLn (ℤ). Ann. Pure Appl. Logic, 161 (2010), 12701279. 56, 58CrossRefGoogle Scholar
Coskey, S.. The classification of torsion-free abelian groups of finite rank up to isomorphism and up to quasi-isomorphism. Trans. Amer. Math. Soc., 364(1) (2012), 175-194. 58CrossRefGoogle Scholar
Coskey, S.. Ioana’s superrigidity theorem and orbit equivalence relations. ISRN Algebra, Art. ID 387540, 8pp. 56, 58Google Scholar
Coskey, S. and Schneider, S.. Cardinal characteristics and countable Borel equivalence relations. Math. Logic Q., 63(3–4) (2017), 211227. 67CrossRefGoogle Scholar
Cotton, M.. Abelian group actions and hypersmooth equivalence relations. Ann. Pure Appl. Logic, 173(8) (2022), 103122. 70CrossRefGoogle Scholar
Day, A.R. and Marks, A.S.. On a question of Slaman and Steel. ArXiv: 2004.00174. 80Google Scholar
Dellacherie, C. and Meyer, P.-A.. Théorie Discrète du Potentiel, Hermann, 1983. 91Google Scholar
de Rancourt, N. and Miller, B.D.. The Feldman–Moore, Glimm–Effros, and Lusin–Novikov theorems over quotients. J. Symb. Logic, to appear. ArXiv:2105.05374. 20Google Scholar
de Rancourt, N. and Miller, B.D.. A dichotomy for countable unions of smooth equivalence relations. J. Symb. Logic, to appear. ArXiv:2105.05362. 20, 33Google Scholar
Ding, L. and Gao, S.. Non-archimedean Polish groups and their actions. Adv. Math., 307 (2017), 312343. 70CrossRefGoogle Scholar
Ditzen, A.. Definable Equivalence Relations on Polish Spaces. Ph.D. Thesis, Caltech, 1992. 37, 39, 41, 44, 45Google Scholar
Dougherty, R., Jackson, S. and Kechris, A.S.. The structure of hyperfinite Borel equivalence relations. Trans. Amer. Math. Soc., 341(1) (1994), 193225. xi, 12, 13, 15, 16, 36, 40, 41, 49, 59, 60, 61, 62, 63, 66, 72, 77, 107, 108CrossRefGoogle Scholar
Dougherty, R. and Kechris, A.S.. How many Turing degrees are there? Contemp. Math., 257 (2000), 8394. 112CrossRefGoogle Scholar
Dye, H.A.. On groups of measure preserving transformations, I. Amer. J. Math., 81 (1959), 119159; II. Amer. J. Math., 85 (1963), 551–576. 61CrossRefGoogle Scholar
Effros, E.G.. Transformation groups and C∗-algebras. Ann. of Math., 81(2) (1965), 3855. 36, 40, 48CrossRefGoogle Scholar
Effros, E.G.. Polish transformation groups and classification problems. In General Topology and Modern Analysis, Academic Press, 1980, 217227. 36, 40, 48Google Scholar
Elayavalli, S.K., Oyakawa, K., Shinko, F. and Spaas, P.. Hyperfiniteness for group actions on trees, ArXiv:2307.10964. 73Google Scholar
Elek, G.. Finite graphs and amenability. J. Funct. Anal., 263 (2012), 25932614 89, 90CrossRefGoogle Scholar
Epstein, I. and Hjorth, G.. Rigidity and equivalence relations with infinitely many ends. logic.ucla.edu/greg/research.html. 83Google Scholar
Epstein, I. and Tsankov, T.. Modular actions and amenable representations. Trans. Amer. Math. Soc., 362(2) (2010), 603621. 54, 102, 103CrossRefGoogle Scholar
Farrell, R.H.. Representation of invariant measures. Ill. J. Math., 6 (1962), 447467. 38Google Scholar
Feldman, J., Hahn, P. and Moore, C.C.. Orbit structure and countable sections for actions of continuous groups. Adv. Math., 26 (1979), 186230. 25, 28Google Scholar
Feldman, J. and Moore, C.C.. Ergodic equivalence relations and von Neumann algebras, I. Trans. Amer. Math. Soc., 234 (1977), 289324. 5, 68CrossRefGoogle Scholar
Forrest, P.. On the virtual groups defined by ergodic actions of ℝn and ℤn. Adv. Math., 14 (1974), 271308. 25CrossRefGoogle Scholar
Frisch, J., Kechris, A.S. and Shinko, F.. Lifts of Borel actions on quotient spaces. Israel J. Math., 251 (2022), 379421. 20, 21, 22CrossRefGoogle Scholar
Frisch, J.R., Kechris, A.S., Shinko, F. and Vidnyánszky, Z.. Realizations of countable Borel equivalence relations. ArXiv:2109.12486. 14, 63, 64, 71, 117, 121, 131, 132, 133, 134, 135, 136Google Scholar
Frisch, J. and Shinko, F.. Quotients by countable subgroups are hyperfinite. Groups Geom. Dyn., 17(3) (2023), 985992. 74CrossRefGoogle Scholar
Furman, A.. Orbit equivalence rigidity. Ann. of Math., 150 (1999), 10831108. 95, 109CrossRefGoogle Scholar
Gaboriau, D.. Coût des relations d’équivalence et des groupes. Invent. Math., 139 (2000), 4198. 6, 93, 97, 100CrossRefGoogle Scholar
Gaboriau, D.. Invariantes 2 de relations d’équivalence et des groupes. Publ. Math. Inst. Hautes Études Sci., 95 (2002), 93150. 106CrossRefGoogle Scholar
Gaboriau, D. and Lyons, R.. A measurable-group-theoretic solution to von Neumann’s problem. Invent. Math., 177(3) (2009), 533540. 79CrossRefGoogle Scholar
Gao, S.. The action of SL2 (ℤ) on the subsets of ℤ2. Proc. Amer. Math. Soc., 129(5) (2000), 15071512. 114CrossRefGoogle Scholar
Gao, S.. Coding subset shift by subgroup conjugacy. Bull. London Math. Soc., 132(6) (2000), 16531657. 113Google Scholar
Gao, S.. Some applications of the Adams–Kechris technique. Proc. Amer. Math. Soc., 130(3) (2002), 863874. 52CrossRefGoogle Scholar
Gao, S. and Hill, A.. Topological isomorphism for rank-1 systems. J. Anal. Math., 128(1) (2016), 149. 74CrossRefGoogle Scholar
Gao, S. and Jackson, S.. Countable abelian groups actions and hyperfinite equivalence relations. Invent. Math., 201 (2015), 309383. 70, 71CrossRefGoogle Scholar
Gao, S., Jackson, S., Krohne, E. and Seward, B.. Forcing constructions and countable Borel equivalence relations. J. Symb. Logic, 87(3) (2022), 873893. 11CrossRefGoogle Scholar
Gao, S., Jackson, S. and Seward, B.. A coloring property for countable groups. Math. Proc. Cambridge Phil. Soc., 147(3) (2009), 579592. 11CrossRefGoogle Scholar
Gao, S., Jackson, S. and Seward, B.. Group colorings and Bernoulli subflows. Mem. Amer. Math. Soc., 241(1141) (2016). 74, 114Google Scholar
Gao, S. and Kechris, A.S.. On the classification of Polish metric spaces up to isometry. Mem. Amer. Math. Soc., 161(766) (2016). 32, 74, 115Google Scholar
Giordano, T. and de la Harpe, P.. Moyennabilité des groupes dénombrables et actions sur les espaces de Cantor. C.R. Acad. Sci. Paris Sér. I Math., 324(11) (1997), 12551258. 134CrossRefGoogle Scholar
Giordano, T., Putnam, I. and Skau, C.. Affable equivalent relations and orbit structure of Cantor dynamical systems. Ergodic Theory Dynam. Systems, 23 (2004), 441475. 6CrossRefGoogle Scholar
Glasner, E. and Weiss, B.. On the interplay between measurable and topological dynamics. In Handbook of Dynamical Systems, Vol. 1B. Hasselblatt, B. and Katok, A. (eds). Elsevier, 2006, 597648. 61Google Scholar
Grebik, J.. σ-lacunary actions of Polish groups. Proc. Amer. Math. Soc., 148(8) (2020), 35833589. 27, 33, 70CrossRefGoogle Scholar
Grebik, J.. Borel equivalence relations induced by actions of tsi Polish groups. ArXiv:2107.14439. 33Google Scholar
Hajian, A.B. and Kakutani, S.. Weakly wandering sets and invariant measures. Trans. Amer. Math. Soc., 110 (1964), 136151. 41CrossRefGoogle Scholar
Halbäck, A., Malicki, M. and Tsankov, T.. Continuous logic and Borel equivalence relations. J. Symb. Logic, 88(4), 17251752. 25CrossRefGoogle Scholar
Harrington, L.A., Kechris, A.S. and Louveau, A.. A Glimm–Effros dichotomy for Borel equivalence relations. J. Amer. Math. Soc., 3(4) (1990), 903928. 30, 36, 48CrossRefGoogle Scholar
Higuchi, K. and Lutz, P.. A note on a question of Sacks: It is harder to embed height-three partial orders than to embed height-two partial orders. ArXiv: 2309.01876. 116Google Scholar
Hjorth, G.. Around nonclassifiability for countable torsion free abelian groups. In Abelian Groups and Modules, Dublin, 1998, Birkhauser, 1999, 269292. 57, 100Google Scholar
Hjorth, G.. Actions by the classical Banach spaces. J. Symb. Logic, 65(1) (2000), 392420. 33CrossRefGoogle Scholar
Hjorth, G.. A converse to Dye’s theorem. Trans. Amer. Math. Soc., 357(8) (2005), 30833103. 54, 101, 102CrossRefGoogle Scholar
Hjorth, G.. Bi-Borel reducibility of essentially countable Borel equivalence relations. J. Symb. Logic, 70(3) (2005), 979992. 23, 96CrossRefGoogle Scholar
Hjorth, G.. A dichotomy for being essentially countable. Contemp. Math., 380 (2005), 109127. 24, 25CrossRefGoogle Scholar
Hjorth, G.. The Furstenberg lemma characterizes amenability. Proc. Amer. Math. Soc., 134(10) (2006), 30613069. 90CrossRefGoogle Scholar
Hjorth, G.. A lemma for cost attained. Ann. Pure Appl. Logic, 143(1–3) (2006), 87102. 95CrossRefGoogle Scholar
Hjorth, G.. Borel equivalence relations which are highly unfree. J. Symb. Logic, 73(4) (2008), 12711277. 109CrossRefGoogle Scholar
Hjorth, G.. Non-treeability for product group actions. Israel J. Math., 163 (2008), 383409. 100CrossRefGoogle Scholar
Hjorth, G.. Selection theorems and treeability. Proc. Amer. Math. Soc., 136(10) (2008), 36473653. 29CrossRefGoogle Scholar
Hjorth, G.. Two generated groups are universal. logic.ucla.edu/greg/research.html. 113Google Scholar
Hjorth, G.. Treeable equivalence relations. J. Math. Logic, 12(1) (2012), 1250003, 21pp. 17, 54, 103CrossRefGoogle Scholar
Hjorth, G. and Kechris, A.S.. Borel equivalence relations and classifications of countable models. Ann. Pure Appl. Logic, 82 (1996), 221272. 30, 31, 40, 65, 94, 98, 100CrossRefGoogle Scholar
Hjorth, G. and Kechris, A.S.. The complexity of the classification of Riemann surfaces and complex manifolds. Ill. J. Math., 44(1) (2000), 104137. 26, 82, 83, 114Google Scholar
Hjorth, G. and Kechris, A.S.. Recent developments in the theory of Borel reducibility. Fund. Math., 170 (2001), 2152. 33, 75CrossRefGoogle Scholar
Hjorth, G. and Kechris, A.S.. Rigidity theorems for actions of product groups and countable Borel equivalence relations. Mem. Amer. Math. Soc., 833 (2005). 16, 17, 51, 54, 57, 58, 101, 106Google Scholar
Hjorth, G., Kechris, A.S. and Louveau, A.. Borel equivalence relations induced by actions of the symmetric group. Ann. Pure Appl. Logic, 92 (1998), 63112. 30CrossRefGoogle Scholar
Hochman, M.. Every Borel automorphism without finite invariant measure admits a two-set generator. J. Eur. Math. Soc., 21(1) (2019), 271317. 117CrossRefGoogle Scholar
Holshouser, J. and Jackson, S.. Partition properties for hyperfinite quotients. Preprint. 77Google Scholar
Huang, S.J., Sabok, M. and Shinko, F.. Hyperfiniteness of boundary actions of cubulated hyperbolic groups. Ergodic Theory Dynam. Systems, 40(9) (2020), 24532466. 73CrossRefGoogle Scholar
Ioana, A.. Relative property (T) for the subequivalence relations induced by the action of SL2 (ℤ) on . Adv. Math., 224(4) (2010), 15891617. 103CrossRefGoogle Scholar
Ioana, A.. Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap. J. Eur. Math. Soc., 18(12) (2016), 27332784. 56, 96, 105CrossRefGoogle Scholar
Ioana, A.. Rigidity for von Neumann algebras. In Proc. of ICM, Rio de Janeiro, 2018, Vol. III: Invited Lectures, World Scientific (2018), 16391672. xiiGoogle Scholar
Ioana, A.. Compact actions whose orbit equivalence relations are not profinite. Adv. Math., 354 (2019), 106753, 19pp. 103CrossRefGoogle Scholar
Iyer, S. and Shinko, F.. The generic action of the free group on Cantor space. In preparation. 136Google Scholar
Jackson, S., Kechris, A.S. and Louveau, A.. Countable Borel equivalence relations. J. Math. Logic, 2(1) (2002), 180. xi, 6, 50, 62, 66, 69, 70, 73, 81, 82, 83, 84, 86, 87, 90, 91, 94, 95, 97, 98, 99, 101, 107, 108, 110, 113, 116, 117, 118CrossRefGoogle Scholar
Kaimanovich, V.A.. Amenability, hyperfiniteness, and isoperimetric inequalities, C. R. Acad. Sci. Paris Ser. I Math., 325(9) (1997), 9991004. 81, 89, 90CrossRefGoogle Scholar
Kanovei, V.. Borel Equivalence Relations, American Mathematical Society, 2008. 24, 33CrossRefGoogle Scholar
Kanovei, V., Sabok, M. and Zapletal, J.. Canonical Ramsey Theory on Polish Spaces, Cambridge University Press, 2013. 34CrossRefGoogle Scholar
Katznelson, Y. and Weiss, B.. The classification of non-singular actions, revisited. Ergodic Theory Dynam. Systems, 11(2) (1991), 333348. 62CrossRefGoogle Scholar
Kaya, B.. The complexity of the topological conjugacy problem for Toeplitz subshifts. Israel J. Math., 220 (2017), 873897. 74CrossRefGoogle Scholar
Kechris, A.S.. Amenable equivalence relations and Turing degrees. J. Symb. Logic, 56(1) (1991), 182194. 87CrossRefGoogle Scholar
Kechris, A.S.. Countable sections for locally compact group actions. Ergodic Theory Dynam. Systems, 12 (1992), 283295. 24, 25, 28, 29CrossRefGoogle Scholar
Kechris, A.S.. Amenable versus hyperfinite Borel equivalence relations. J. Symb. Logic, 58(3) (1993), 894907. 91CrossRefGoogle Scholar
Kechris, A.S.. Countable sections for locally compact group actions, II. Proc. Amer. Math. Soc., 120(1) (1994), 241247. 26, 62CrossRefGoogle Scholar
Kechris, A.S.. Lectures on definable group actions and equivalence relations. Preprint, 1994. 37, 39, 41, 42, 45, 64, 65, 68Google Scholar
Kechris, A.S.. Classical Descriptive Set Theory, Springer, 1995. 5, 25, 28, 31, 38, 42, 77CrossRefGoogle Scholar
Kechris, A.S.. On the classification problem for rank 2 torsion-free abelian groups. J. London Math. Soc., 62(2) (2000), 437450. 100, 101CrossRefGoogle Scholar
Kechris, A.S.. Descriptive dynamics. In Descriptive Set Theory and Dynamical Systems, London Mathematical Society Lecture Note Series, 277, Cambridge University Press, 2000, 231258. 25CrossRefGoogle Scholar
Kechris, A.S.. Unitary representations and modular actions. J. Math. Sci., 140(3) (2007), 398425. 54, 101, 102CrossRefGoogle Scholar
Kechris, A.S.. Global Aspects of Ergodic Group Actions, American Mathematical Society, 2010. 17, 54, 109CrossRefGoogle Scholar
Kechris, A.S.. The spaces of measure preserving equivalence relations and graphs. pma.caltech.edu/people/alexander-kechris. 5, 106Google Scholar
Kechris, A.S.. Quasi-invariant measures for continuous group actions. Contemp. Math., 752 (2020), 113120. 42CrossRefGoogle Scholar
Kechris, A.S. and Louveau, A.. The classification of hypersmooth Borel equivalence relations. J. Amer. Math. Soc., 10(1) (1997), 215242. 75CrossRefGoogle Scholar
Kechris, A.S. and Macdonald, H.L.. Borel equivalence relations and cardinal algebras. Fund. Math., 235 (2016), 183198. 24, 124, 125Google Scholar
Kechris, A.S., Malicki, M., Panagiotopoulos, A. and Zielinski, Z.. On Polish groups admitting non-essentially countable actions. Ergodic Theory Dynam. Systems, 42 (2022), 180194. 28CrossRefGoogle Scholar
Kechris, A.S. and Marks, A.S.. Descriptive graph combinatorics. pma.caltech.edu/people/alexander-kechris. xiiGoogle Scholar
Kechris, A.S. and Miller, B.D.. Topics in Orbit Equivalence, Springer, 2004. xii, 11, 13, 14, 38, 43, 44, 61, 65, 79, 81, 93, 94, 99CrossRefGoogle Scholar
Kechris, A.S. and Miller, B.D.. Means on equivalence relations. Israel J. Math., 163 (2008), 241262. 9CrossRefGoogle Scholar
Kechris, A.S., Solecki, S. and Todorcevic, S.. Borel chromatic numbers. Adv. Math., 141 (1999), 144. 5, 63, 126CrossRefGoogle Scholar
Kechris, A.S. and Wolman, M.. Ditzen’s effective version of Nadkarni’s Theorem. pma.caltech.edu/people/alexander-kechris. 37, 39Google Scholar
Kerr, D. and Li, H.. Ergodic Theory, Springer, 2016. 90CrossRefGoogle Scholar
Khezeli, A.. Shift-coupling of random rooted graphs and networks. Contemp. Math., 719 (2018), 175211. 45CrossRefGoogle Scholar
Kida, Y. and Tucker-Drob, R.. Inner amenable groupoids and central sequences. Forum Math. Sigma, 8 (2020), Paper No. e29, 84 pp. 90CrossRefGoogle Scholar
Krieger, W.. On non-singular transformations of a measure space, I. Z. Wahrsch. Verw. Gebiete, 11 (1969), 8397; II. ibid, 11 (1969), 98–119. 62CrossRefGoogle Scholar
Krupinski, K., Pillay, A. and Solecki, S.. Borel equivalence relations and Lascar strong types. J. Math. Logic, 13(2) (2013), 1350008, 37 pp. 74CrossRefGoogle Scholar
Kyed, D., Petersen, H.D. and Vaes, S.. L2-Betti numbers of locally compact groups and their cross section equivalence relations. Trans. Amer. Math. Soc., 367(7) (2015), 49174956. 26, 37CrossRefGoogle Scholar
Larson, P.B.. The filter dichotomy and medial limits. J. Math. Logic, 9(2) (2009), 159165. 91CrossRefGoogle Scholar
Lecomte, D.. On the complexity of Borel equivalence relations with some countability property. Trans. Amer. Math. Soc., 373(3), (2020), 18451883. 22CrossRefGoogle Scholar
Levitt, G.. On the cost of generating an equivalence relation. Ergodic Theory Dynam. Systems, 15 (1995), 11731181. 93CrossRefGoogle Scholar
Lopez-Escobar, E.G.K.. An interpolation theorem for denumerably long formulas. Fund. Math., 57 (1965), 253272. 127CrossRefGoogle Scholar
Louveau, A. and Mokobodzki, G.. On measures ergodic with respect to an analytic equivalence relation. Trans. Amer. Math. Soc., 349(12) (1997), 48154823. 45CrossRefGoogle Scholar
Lupini, M.. Polish groupoids and functorial complexity. Trans. Amer. Math. Soc., 369(9) (2017), 66836723. 95CrossRefGoogle Scholar
Lutz, P. and Siskind, B.. Part 1 of Martin’s conjecture for order preserving and measure preserving functions. ArXiv: 2305.19646. 116Google Scholar
Malicki, M.. Abelian pro-countable groups and orbit equivalence relations. Fund. Math., 233(1) (2016), 8399. 28Google Scholar
Marker, D.. The Borel complexity of isomorphism for theories with many types. Notre Dame J. Formal Logic, 48(1) (2007), 9397. 32CrossRefGoogle Scholar
Marks, A.S.. A determinacy approach to Borel combinatorics. J. Amer. Math. Soc., 29(2) (2016), 579600. 10CrossRefGoogle Scholar
Marks, A.S.. The universality of polynomial time Turing equivalence. Math. Structures Comput. Sci., 28(3) (2018), 448456. 112CrossRefGoogle Scholar
Marks, A.S.. Uniformity, universality, and computability theory. J. Math. Logic, 17 (2017), 1750003, 50 pp. 11, 111, 112, 119, 120, 121, 126, 127, 129CrossRefGoogle Scholar
Marks, A.S.. A short proof of the Connes-Feldman–Weiss theorem. math.berkeley.edu/~marks/. 68, 78, 90Google Scholar
Marks, A.S., Slaman, T.A. and Steel, J.R.. Martin’s Conjecture, Arithmetic Equivalence and Countable Borel Equivalence Relations. In Lecture Notes in Logic, Assoc. Symb. Logic, 43, 2016, 493519. 110, 111, 119Google Scholar
Marquis, J.. On geodesic ray bundles in buildings. Geom. Dedicata, 202 (2019), 2743. 73CrossRefGoogle Scholar
Marquis, J. and Sabok, M.. Hyperfiniteness of boundary actions of hyperbolic groups. Math. Ann., 377(3–4) (2020), 11291153. 73CrossRefGoogle Scholar
Mercer, R.. The full group of a countable measurable equivalence relation. Proc. Amer. Math. Soc., 117(2) (1993), 323333. 18CrossRefGoogle Scholar
Miller, B.D.. A bireducibility lemma. Item 4 under Unpublished in glimmeffros.github.io. 15Google Scholar
Miller, B.D.. Borel dynamics. preprint, 2007. 18, 44Google Scholar
Miller, B.D.. A generalized marker lemma. Item 2 under Unpublished in glimmeffros.github.io. 9Google Scholar
Miller, B.D.. The classification of finite Borel equivalence relations on 2/E0. Item 12 under Unpublished in glimmeffros.github.io. 76Google Scholar
Miller, B.D., Full groups, classification, and equivalence relations. Ph.D. Thesis, U.C. Berkeley, 2004. Item 2 under Publications in glimmeffros. github.io. 16, 18, 20Google Scholar
Miller, B.D., Borel equivalence relations and everywhere faithful actions of free products. Item 8 under Unpublished in glimmeffros.github.io. 109Google Scholar
Miller, B.D.. On the existence of invariant probability measures for Borel actions of countable semigroups. Item 9 under Unpublished in glimmeffros. github.io. 14Google Scholar
Miller, B.D.. The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures. Ergodic Theory Dynam. Systems, 28 (2008), 15991613. 44CrossRefGoogle Scholar
Miller, B.D.. The existence of measures of a given cocycle, II: probability measures. Ergodic Theory Dynam. Systems, 28 (2008), 16151633. 44, 45CrossRefGoogle Scholar
Miller, B.D.. Ends of graphed equivalence relations, I. Israel J. Math., 169 (2009), 375392 74CrossRefGoogle Scholar
Miller, B.D.. Incomparable treeable equivalence relations. J. Math. Logic, 12(1) (2012), 1250004, 11pp. 17, 54, 103CrossRefGoogle Scholar
Miller, B.D.. On the existence of cocycle-invariant probability measures. Ergodic Theory Dynam. Systems, 40(11) (2020), 31503168. 44CrossRefGoogle Scholar
Miller, B.D.. A generalization of the -dichotomy and a strengthening of the -dichotomy. J. Math. Logic, 22(1) (2022), Paper No. 2150028, 19pp. 27, 33CrossRefGoogle Scholar
Miller, B.D.. The existence of invariant measures. Item 3 under Seminars in glimmeffros.github.io. 40, 44Google Scholar
Miller, B.D.. Reducibility of countable equivalence relations. Item 4 under Seminars in glimmeffros.github.io. 103Google Scholar
Miller, B.D.. Compositions of periodic automorphisms. Item 2 under Recent in glimmeffros.github.io. 18Google Scholar
Miller, B.D.. Essential values of cocycles and the Borel structure of R/Q. Item 3 under Recent in glimmeffros.github.io. 76Google Scholar
Miller, B.D.. A first-order characterization of the existence of invariant measures. Item 4 under Recent in glimmeffros.github.io. 18Google Scholar
Miller, B.D.. A generalization of the Dye–Krieger theorem. Item 23 under Unpublished in glimmeffros.github.io. 68Google Scholar
Miller, B.D. and Rosendal, C.. Isomorphism of Borel full groups. Proc. Amer. Math. Soc., 135(2) (2007), 517522. 17, 18CrossRefGoogle Scholar
Miller, B.D. and Rosendal, C.. Descriptive Kakutani equivalence. J. Eur. Math. Soc., 12(1) (2010), 179219. 62CrossRefGoogle Scholar
Moore, J.T.. A brief introduction to amenable equivalence relations. Contemp. Math., 752 (2020), 153164. 66, 91, 92CrossRefGoogle Scholar
Mycielski, J.. (untitled) Amer. Math. Monthly, 82(3) (1975), 308309. 72CrossRefGoogle Scholar
Nadkarni, M.G.. Descriptive ergodic theory. Contemp. Math., 94 (1989), 191209. 12, 13CrossRefGoogle Scholar
Nadkarni, M.G.. On the existence of a finite invariant measure. Proc. Indian Acad. Sci. Math. Sci, 100 (1991), 203220. 12, 37CrossRefGoogle Scholar
Nadkarni, M.G.. Basic Ergodic Theory, 3rd edition, Birkhäuser, 2013. 41CrossRefGoogle Scholar
Naryshkin, P. and Vaccaro, A.. Hyperfiniteness and Borel asymptotic dimension of boundary actions of hyperbolic groups. ArXiv: 2306, 02056. 73Google Scholar
Nebbia, C.. Amenability and Kunze–Stein property for groups acting on a tree. Pacific J. Math., 135 (1988), 371380. 87CrossRefGoogle Scholar
Ornstein, D. and Weiss, B.. Ergodic theory and amenable group actions, I: The Rohlin lemma. Bull. Amer. Math. Soc. (NS), 2 (1980), 161164. 69CrossRefGoogle Scholar
Oyakawa, K.. Hyperfiniteness of boundary actions of acylindrically hyperbolic groups. ArXiv: 2307.09790. 73Google Scholar
Panagiotopoulos, A. and Wang, A.. Every CBER is smooth below the Carlson–Simpson generic partition. ArXiv:2206.14224. 34Google Scholar
Paterson, A.L.T.. Amenability, American Mathematical Society, 1988. 81CrossRefGoogle Scholar
Pikhurko, O.. Borel combinatorics of locally finite graphs. In Surveys in Combinatorics, London Mathematical Society Lecture Note Series, 470, 2021, 267319. xiiGoogle Scholar
Popa, S.. Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups. Invent. Math., 170(2) (2007), 243295. 54CrossRefGoogle Scholar
Przytycki, P. and Sabok, M.. Unicorn paths and hyperfiniteness for the mapping class group. Forum Math. Sigma, 9 (2021), Paper No. e36, 10pp. 73CrossRefGoogle Scholar
Rosendal, C.. On the non-existence of certain group topologies. Fund. Math., 183(3) (2005), 213228. 18CrossRefGoogle Scholar
Quorning, V.. Superrigidity, Profinite Actions and Borel Reducibility. Master’s Thesis, University of Copenhagen, 2015. 58Google Scholar
Ramsey, A.. Topologies on measure groupoids. J. Funct. Anal., 47 (1982), 314343. 28CrossRefGoogle Scholar
Rudolph, D.. A two-valued step coding for ergodic flows. Math. Z., 150(3) (1976), 201220. 27CrossRefGoogle Scholar
Sabok, M. and Tsankov, T.. On the complexity of topological conjugacy of Toeplitz subshifts. Israel J. Math., 220 (2017), 583603. 74CrossRefGoogle Scholar
Schneider, S.. Simultaneous reducibility of pairs of Borel equivalence relations. ArXiv:1310.8028. 48Google Scholar
Schneider, S. and Seward, B.. Locally nilpotent groups and hyperfinite equivalence relations. ArXiv: 1308.5853. 71Google Scholar
Segal, M.. Hyperfiniteness. Ph.D. Thesis, Hebrew University of Jerusalem, 1997. 77, 78Google Scholar
Seward, B. and Tucker-Drob, R.D.. Borel structurability of the 2-shift of a countable group. Ann. Pure Appl. Logic, 167(1) (2016), 121. 98, 107CrossRefGoogle Scholar
Shinko, F.. Equidecomposition in cardinal algebras. Fund. Math., 253(2) (2021), 197204. 46CrossRefGoogle Scholar
Silver, J.H.. Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Ann. Math. Logic, 18 (1980), 128. 47CrossRefGoogle Scholar
Slaman, T.A. and Steel, J.R.. Definable functions on degrees. In Cabal Seminar 81–85, Lecture Notes in Mathematics, 1333, 1988, Springer-Verlag, 3755. 9, 50, 59, 80, 101CrossRefGoogle Scholar
Slutsky, K.. Lebesgue orbit equivalence of multidimensional Borel flows: a picturebook of tilings. Ergodic Theory Dynam. Systems, 37(6) (2017), 19661996. 26, 27, 37, 62CrossRefGoogle Scholar
Slutsky, K.. Regular cross sections of Borel flows. J. Eur. Math. Soc., 21(7) (2019), 19852050. 13, 27, 62CrossRefGoogle Scholar
Slutsky, K.. Cross sections of Borel flows with restrictions on the distance set. ArXiv: 1604.02215. 27Google Scholar
Slutsky, K.. On time change equivalence of Borel flows. Fund. Math., 247(1) (2019), 124. 27, 62CrossRefGoogle Scholar
Slutsky, K.. Countable Borel equivalence relations. Preprint, kslutsky.com. 13, 37, 39, 60, 61Google Scholar
Smythe, I.B.. Equivalence of generics. Arch. Math. Logic, 61(5–6) (2022), 795812. 66CrossRefGoogle Scholar
Solecki, S.. Actions of non-compact and non-locally compact groups. J. Symb. Logic, 65(4) (2000), 18811894. 27, 28CrossRefGoogle Scholar
Sullivan, D., Weiss, B. and Wright, J.D.M.. Generic dynamics and monotone complete C∗-algebras. Trans. Amer. Math. Soc., 295(20) (1986), 795809. 65Google Scholar
Tarski, A.. Cardinal Algebras, Oxford University Press, 1949. 123, 124, 125Google Scholar
Thomas, S.. The classification problem for p-local torsion-free abelian groups of finite rank. Preprint, sites.math.rutgers.edu/~sthomas/papers.html. 56Google Scholar
Thomas, S.. Some applications of superrigidity to Borel equivalence relations. In DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 58 (2002), American Mathematical Society, 129134. 16Google Scholar
Thomas, S.. The classification problem for torsion-free abelian groups of finite rank. J. Amer. Math. Soc., 16(1) (2003), 233258. 57, 58, 98, 100CrossRefGoogle Scholar
Thomas, S.. Superrigidity and countable Borel equivalence relations. Ann. Pure Appl. Logic, 120 (2003), 237262. 56, 58CrossRefGoogle Scholar
Thomas, S.. Borel superrigidity and the classification problem for the torsion-free abelian groups of finite rank. In Proc. ICM, European Mathematical Society, 2006, 93116. 57, 58Google Scholar
Thomas, S.. Property (τ) and countable Borel equivalence relations. J. Math. Logic, 7(1) (2007), 134. 56CrossRefGoogle Scholar
Thomas, S.. The classification problem for finite rank Butler groups. In Models, Modules and Abelian Groups, Walter de Gruyter, 2008, 329338. 73CrossRefGoogle Scholar
Thomas, S.. On the complexity of the quasi-isometry and virtual isomorphism problems for finitely generated groups. Groups Geom. Dyn., 2 (2008), 281307. 32, 115CrossRefGoogle Scholar
Thomas, S.. Continuous versus Borel reductions. Arch. Math. Logic, 48 (2009), 761770. 71CrossRefGoogle Scholar
Thomas, S.. The commensurability relation for finitely generated groups. J. Group Theory, 12 (2009), 901909. 113CrossRefGoogle Scholar
Thomas, S.. Martin’s conjecture and strong ergodicity. Arch. Math. Logic, 48 (2009), 749759. 67, 118, 119CrossRefGoogle Scholar
Thomas, S.. Popa superrigidity and countable Borel equivalence relations. Ann. Pure Appl. Logic, 158 (2009), 175189. 16, 17, 51, 108, 109, 110, 119CrossRefGoogle Scholar
Thomas, S.. The classification problem for S-local torsion-free abelian groups of finite rank. Adv. Math., 226 (2011), 36993723. 58CrossRefGoogle Scholar
Thomas, S.. Universal Borel actions of countable groups. Groups Geom. Dyn., 6 (2012), 389407. 107, 109, 113, 116, 117, 119CrossRefGoogle Scholar
Thomas, S.. On the E0-extensions of countable Borel equivalence relations. Preprint, sites.math.rutgers.edu/~sthomas/papers.html. 49Google Scholar
Thomas, S. and Schneider, S.. Countable Borel equivalence relations. In Appalachian Set Theory, 2006–2012, London Mathematical Society Lecture Note Series, 406, 2013, 2562. 58Google Scholar
Thomas, S. and Velickovic, B.. On the complexity of the isomorphism relation for finitely generated groups. J. Algebra, 217 (1999), 352373. 113CrossRefGoogle Scholar
Thomas, S. and Velickovic, B.. On the complexity of the isomorphism relation for fields of finite transcendence degree. J. Pure Appl. Alg., 159 (2001), 347363. 113CrossRefGoogle Scholar
Thomas, S. and Williams, J.. The bi-embeddability relation for finitely generated groups, II. Arch. Math. Logic, 55 (2016), 483500. 118CrossRefGoogle Scholar
Thompson, A.. A metamathematical condition equivalent to the existence of a complete left invariant metric. J. Symb. Logic, 71(4) (2006), 11081124. 27CrossRefGoogle Scholar
Thorisson, H.. Transforming random elements and shifting random fields. Ann. Probab., 24(4) (1996), 20572064. 45CrossRefGoogle Scholar
Tserunyan, A.. Hjorth’s proof of the embeddability of hyperfinite equivalence relations into E0. Preprint, math.mcgill.ca/atserunyan/research.html. 60Google Scholar
Tserunyan, A.. Segal’s effective witness to measure-hyperfiniteness. Preprint, math.mcgill.ca/atserunyan/research.html. 68, 78Google Scholar
Tserunyan, A.. Finite Generators for Countable Group Actions; Finite Index Pairs of Equivalence Relations; Complexity Measures for Recursive Programs. Ph.D. Thesis, UCLA, 2013. math.mcgill.ca/atserunyan/research.html. 96Google Scholar
Tserunyan, A.. Finite generators for countable group actions in the Borel and Baire category settings. Adv. Math., 269 (2015), 585646. 117CrossRefGoogle Scholar
Tserunyan, A.. A descriptive construction of trees and Stallings’ theorem. Contemp. Math., 752, 191207. 99CrossRefGoogle Scholar
Tserunyan, A. and Tucker-Drob, R.. The Radon–Nikodym topography of amenable equivalence relations in an acyclic graph. In preparation. 97Google Scholar
Varadarajan, V.S.. Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc., 109 (1963), 191220. 38CrossRefGoogle Scholar
Wagh, V.M.. A descriptive version of Ambrose’s representation theorem for flows. Proc. Indian Acad. Sci. Math. Sci., 98 (1988), 101108. 25CrossRefGoogle Scholar
Weiss, B.. Measurable dynamics. Contemp. Math., 26 (1984), 395421. 36, 40, 48, 59, 69CrossRefGoogle Scholar
Weiss, B.. Countable generators in dynamics – universal minimal models. Contemp. Math., 94 (1989), 321326. 117CrossRefGoogle Scholar
Williams, J.. Universal countable Borel quasi-orders. J. Symb. Logic, 79 (2014), 928954. 115, 116Google Scholar
Williams, J.. Isomorphism of finitely generated solvable groups is weakly universal. J. Pure Appl. Algebra, 219(5) (2015), 16391644. 118CrossRefGoogle Scholar
Zimmer, R.J.. Hyperfinite factors and amenable ergodic actions. Invent. Math., 41 (1977), 2331. 88CrossRefGoogle Scholar
Zimmer, R.J.. Ergodic Theory and Semisimple Groups. Birkhäuser, 1984. xii, 52, 55, 88CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Alexander S. Kechris, California Institute of Technology
  • Book: The Theory of Countable Borel Equivalence Relations
  • Online publication: 07 November 2024
  • Chapter DOI: https://doi.org/10.1017/9781009562256.018
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Alexander S. Kechris, California Institute of Technology
  • Book: The Theory of Countable Borel Equivalence Relations
  • Online publication: 07 November 2024
  • Chapter DOI: https://doi.org/10.1017/9781009562256.018
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Alexander S. Kechris, California Institute of Technology
  • Book: The Theory of Countable Borel Equivalence Relations
  • Online publication: 07 November 2024
  • Chapter DOI: https://doi.org/10.1017/9781009562256.018
Available formats
×