Bull. Amer. Math. Soc. 63 (2026), 637-684
DOI: https://doi.org/10.1090/bull/1908
Published electronically: August 13, 2026
Abstract
A general overview of the phenomenon of automatic continuity of homomorphisms between Polish groups is given. In particular, we study variants and improvements of the closed graph theorem, applying these to the problem of continuity of universally measurable homomorphisms as well as to gauge the amount of choice needed to construct discontinuous homomorphisms between Polish groups. Furthermore, we provide a simple proof of automatic continuity in the context of homeomorphism groups of compact manifolds and a complete reworking of automatic continuity theory in the context of isometry groups of highly homogeneous complete metric structures.
- Roger Alperin, Locally compact groups acting on trees, Pacific J. Math. 100 (1982), no. 1, 23–32. MR 661438, DOI 10.2140/pjm.1982.100.23
- T. Banakh, Automatic continuity of measurable homomorphisms on Čech-complete topological groups (2022).
- Itaï Ben Yaacov, Alexander Berenstein, and Julien Melleray, Polish topometric groups, Trans. Amer. Math. Soc. 365 (2013), no. 7, 3877–3897. MR 3042607, DOI 10.1090/S0002-9947-2013-05773-X
- Oleg Bogopolski and Samuel M. Corson, Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups, Math. Ann. 384 (2022), no. 3-4, 1017–1055. MR 4498466, DOI 10.1007/s00208-021-02278-4
- Salomon Bochner and Deane Montgomery, Groups of differentiable and real or complex analytic transformations, Ann. of Math. (2) 46 (1945), 685–694. MR 14102, DOI 10.2307/1969204
- Jan Brzuchowski, Jacek Cichoń, Edward Grzegorek, and Czesław Ryll-Nardzewski, On the existence of nonmeasurable unions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 6, 447–448 (English, with Russian summary). MR 560178
- Lev Bukovský, Any partition into Lebesgue measure zero sets produces a nonmeasurable set, Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 6, 431–435 (English, with Russian summary). MR 560175
- Lei Chen and Kathryn Mann, Structure theorems for actions of homeomorphism groups, Duke Math. J. 172 (2023), no. 5, 915–962. MR 4568050, DOI 10.1215/00127094-2022-0019
- Jens Peter Reus Christensen, Borel structures in groups and semigroups, Math. Scand. 28 (1971), 124–128. MR 308322, DOI 10.7146/math.scand.a-11010
- D. van Dantzig, Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen, Compositio Math. 3 (1936), 408–426 (German). MR 1556954
- Carlos Augusto Di Prisco and Stevo Todorcevic, Perfect-set properties in $L(\mathbf {R})[U]$, Adv. Math. 139 (1998), no. 2, 240–259. MR 1654181, DOI 10.1006/aima.1998.1752
- John D. Dixon, Peter M. Neumann, and Simon Thomas, Subgroups of small index in infinite symmetric groups, Bull. London Math. Soc. 18 (1986), no. 6, 580–586. MR 859950, DOI 10.1112/blms/18.6.580
- R. M. Dudley, Continuity of homomorphisms, Duke Math. J. 28 (1961), 587–594. MR 136676, DOI 10.1215/S0012-7094-61-02859-9
- Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
- David Epstein and Vladimir Markovic, Extending homeomorphisms of the circle to quasiconformal homeomorphisms of the disk, Geom. Topol. 11 (2007), 517–595. MR 2302497, DOI 10.2140/gt.2007.11.517
- I. I. Guran, Topological groups similar to Lindelöf groups, Dokl. Akad. Nauk SSSR 256 (1981), no. 6, 1305–1307 (Russian). MR 606469
- Wilfrid Hodges, Ian Hodkinson, Daniel Lascar, and Saharon Shelah, The small index property for $\omega$-stable $\omega$-categorical structures and for the random graph, J. London Math. Soc. (2) 48 (1993), no. 2, 204–218. MR 1231710, DOI 10.1112/jlms/s2-48.2.204
- Adriane Kaïchouh and François Le Maître, Connected Polish groups with ample generics, Bull. Lond. Math. Soc. 47 (2015), no. 6, 996–1009. MR 3431579, DOI 10.1112/blms/bdv078
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- Alexander S. Kechris, Global aspects of ergodic group actions, Mathematical Surveys and Monographs, vol. 160, American Mathematical Society, Providence, RI, 2010. MR 2583950, DOI 10.1090/surv/160
- Alexander S. Kechris and Christian Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc. (3) 94 (2007), no. 2, 302–350. MR 2308230, DOI 10.1112/plms/pdl007
- Sang-hyun Kim and Thomas Koberda, Structure and regularity of group actions on one-manifolds, Springer Monographs in Mathematics, Springer, Cham, [2021] ©2021. MR 4381312, DOI 10.1007/978-3-030-89006-3
- Aleksandra Kwiatkowska, The group of homeomorphisms of the Cantor set has ample generics, Bull. Lond. Math. Soc. 44 (2012), no. 6, 1132–1146. MR 3007646, DOI 10.1112/blms/bds039
- Paul B. Larson and Jindřich Zapletal, Discontinuous homomorphisms, selectors, and automorphisms of the complex field, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1733–1737. MR 3910437, DOI 10.1090/proc/14338
- Paul B. Larson and Jindřich Zapletal, Geometric set theory, Mathematical Surveys and Monographs, vol. 248, American Mathematical Society, Providence, RI, [2020] ©2020. MR 4249448, DOI 10.1090/surv/248
- Maciej Malicki, Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures, J. Symb. Log. 81 (2016), no. 3, 876–886. MR 3569109, DOI 10.1017/jsl.2015.73
- Maciej Malicki, An example of a non non-archimedean Polish group with ample generics, Proc. Amer. Math. Soc. 144 (2016), no. 8, 3579–3581. MR 3503726, DOI 10.1090/proc/13017
- Kathryn Mann, Automatic continuity for homeomorphism groups and applications, Geom. Topol. 20 (2016), no. 5, 3033–3056. With an appendix by Frédéric Le Roux and Mann. MR 3556355, DOI 10.2140/gt.2016.20.3033
- Nikolay Nikolov and Dan Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171–238. MR 2276769, DOI 10.4007/annals.2007.165.171
- Claude Amboise Rogers (ed.), Analytic sets, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. Lectures delivered at a Conference held at University College, University of London, London, July 16–29, 1978. MR 0608794
- Christian Rosendal, Automatic continuity in homeomorphism groups of compact 2-manifolds, Israel J. Math. 166 (2008), 349–367. MR 2430439, DOI 10.1007/s11856-008-1034-x
- Christian Rosendal, A topological version of the Bergman property, Forum Math. 21 (2009), no. 2, 299–332. MR 2503307, DOI 10.1515/FORUM.2009.014
- Christian Rosendal, Automatic continuity of group homomorphisms, Bull. Symbolic Logic 15 (2009), no. 2, 184–214. MR 2535429, DOI 10.2178/bsl/1243948486
- Christian Rosendal, Finitely approximable groups and actions part II: Generic representations, J. Symbolic Logic 76 (2011), no. 4, 1307–1321. MR 2895397, DOI 10.2178/jsl/1318338851
- Christian Rosendal, Continuity of universally measurable homomorphisms, Forum Math. Pi 7 (2019), e5, 20. MR 3996719, DOI 10.1017/fmp.2019.5
- Christian Rosendal and Sławomir Solecki, Automatic continuity of homomorphisms and fixed points on metric compacta, Israel J. Math. 162 (2007), 349–371. MR 2365867, DOI 10.1007/s11856-007-0102-y
- Marcin Sabok, Automatic continuity for isometry groups, J. Inst. Math. Jussieu 18 (2019), no. 3, 561–590. MR 3936642, DOI 10.1017/s1474748017000135
- Marcin Sabok, Automatic continuity for isometry groups—erratum, J. Inst. Math. Jussieu 21 (2022), no. 6, 2253–2255. MR 4515296, DOI 10.1017/S1474748020000766
- Jan Saxl and John S. Wilson, A note on powers in simple groups, Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 91–94. MR 1443588, DOI 10.1017/S030500419600165X
- Saharon Shelah, Can you take Solovay’s inaccessible away?, Israel J. Math. 48 (1984), no. 1, 1–47. MR 768264, DOI 10.1007/BF02760522
- Konstantin Slutsky, Automatic continuity for homomorphisms into free products, J. Symbolic Logic 78 (2013), no. 4, 1288–1306. MR 3156525, DOI 10.2178/jsl.7804160
- Ernst Specker, Additive Gruppen von Folgen ganzer Zahlen, Portugal. Math. 9 (1950), 131–140 (German). MR 39719
- Sławomir Solecki, Actions of non-compact and non-locally compact Polish groups, J. Symbolic Logic 65 (2000), no. 4, 1881–1894. MR 1812189, DOI 10.2307/2695084
- Sławomir Solecki, Extending partial isometries, Israel J. Math. 150 (2005), 315–331. MR 2255813, DOI 10.1007/BF02762385
- Robert M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92 (1970), 1–56. MR 265151, DOI 10.2307/1970696
- Hugo Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fund. Math. 1 (1920), 93–104., DOI 10.4064/fm-1-1-93-104
- Terence Tao, Hilbert’s fifth problem and related topics, Graduate Studies in Mathematics, vol. 153, American Mathematical Society, Providence, RI, 2014. MR 3237440, DOI 10.1090/gsm/153
- Simon Thomas, Infinite products of finite simple groups. II, J. Group Theory 2 (1999), no. 4, 401–434. MR 1718758, DOI 10.1515/jgth.1999.029
- Simon Thomas and Jindřich Zapletal, On the Steinhaus and Bergman properties for infinite products of finite groups, Confluentes Math. 4 (2012), no. 2, 1250002, 26. MR 2982769, DOI 10.1142/S1793744212500028
- Todor Tsankov, Automatic continuity for the unitary group, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3673–3680. MR 3080189, DOI 10.1090/S0002-9939-2013-11666-7
- P. S. Urysohn, Sur un espace métrique universel, C. R. Acad. Sci. Paris 180 (1925), 803–806.
- P. S. Urysohn, Sur un espace métrique universel, Bull. Sci. Math. 51 (1927), 43–64 and 74–90.
- Vladimir V. Uspenskij, On subgroups of minimal topological groups, Topology Appl. 155 (2008), no. 14, 1580–1606. MR 2435151, DOI 10.1016/j.topol.2008.03.001
- André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 869, Hermann & Cie, Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 5741
- Phillip Wesolek, Conjugacy class conditions in locally compact second countable groups, Proc. Amer. Math. Soc. 144 (2016), no. 1, 399–409. MR 3415606, DOI 10.1090/proc/12645
- Christian Rosendal
- Affiliation: Department of Mathematics, University of Maryland, 4176 Campus Drive - William E. Kirwan Hall, College Park, Maryland 20742-4015, USA
- MR Author ID: 688175
- Email: [email protected]
- Luis Carlos Suarez
- Affiliation: Department of Mathematics, University of Maryland, 4176 Campus Drive - William E. Kirwan Hall, College Park, Maryland 20742-4015, USA
- Email: [email protected]
- Received by editor(s): September 22, 2025
- Received by editor(s) in revised form: March 21, 2026
- Published electronically: August 13, 2026
- Additional Notes: The authors acknowledge support by the U.S. National Science Foundation under Award Numbers DMS-2246986 and DMS-2204849.
- © Copyright 2026
American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 63 (2026), 637-684
- MSC (2020): Primary 22A05; Secondary 03E15, 03E25
- DOI: https://doi.org/10.1090/bull/1908
SubmissionThe AMS uses Centralized Manuscript Processing for initial submissions to AMS journals. The preferred method of submission is to upload a PDF file using the Initial Manuscript Submission form.
An alternate method is to send one copy of the paper to the following address:
Centralized Manuscript Processing
Bulletin of the AMS
201 Charles Street
Providence, Rhode Island
02904-2213 USAIf a paper copy is being forwarded to the AMS, the following information is required with the submission (note that all contact information, particularly email addresses, must be supplied to avoid delay):
- Name of the journal to which the paper is submitted
- Name of the corresponding author
- Contact information including email address and mailing address
- The author should suggest an appropriate Editor to review the paper
No paper that has been previously published, or is being considered for publication elsewhere, should be submitted to the American Mathematical Society, nor may a paper that is under consideration by the American Mathematical Society be submitted elsewhere.
Included with the footnotes to the paper should be the 2020 Mathematics Subject Classification representing the primary and secondary subjects of the article.
The AMS Author Resource Center provides TeX Resources, the AMS Style Guide, the Mathematics Subject Classification, and further useful tools for preparing your manuscript. Journal abbreviations used in bibliographies are available for download.
After AcceptanceOnce a paper has been accepted for publication, authors should follow the online Submission Instructions.
All authors must submit the Consent to Publish. All authors must agree regarding the Copyright Agreement. If the article is to be published in one of the AMS Open Access journals, authors must agree which Creative Commons License will be selected. Authors are encouraged to use AMS-prepared style files in preparing their papers.
Useful ToolsAn abstract is required. It should be a brief technical description of the new material. The AMS offers free worldwide access to the electronic abstracts. No abstracts appear in the printed journal starting in 1998.