Abstract
We prove results about automatic continuity and openness of abstract surjective group homomorphisms \( K\overset{\varphi }{\to }G, \) where G and K belong to a certain class К of topological groups, and where the kernel of φ satisies a certain topological countability condition. Our results apply in particular to the case where G is a semisimple Lie group or a semisimple compact group, and where К is either the class of all locally compact groups or the class of all Polish groups.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. M. Al-Tameemi, R. R. Kallman, The natural semidirect product ℝn ⋊G(n) is algebraically determined, Topology Appl. 199 (2016), 70–83.
A. Borel, J. Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571.
N. Bourbaki, General Topology, Chapters 1–4, Springer, Berlin, 1989. N. Bourbaki, General Topology, Chapters 5–10, Springer, Berlin, 1989.
O. Braun, Uniqueness of Topologies on Compact Connected Groups, Diploma Thesis, Univ. Münster, 2016.
É. Cartan, Sur les représentations linéaires des groupes clos, Comment. Math. Helv. 2 (1930), no. 1, 269–283.
L. Diels, P. A. Dowerk, Invariant automatic continuity for compact connected simple Lie groups, arXiv:1811.04618v1 (2018).
P. A. Dowerk, A. Thom, Bounded normal generation and invariant automatic continuity, Adv. in Math. 346 (2019), 124–169.
J. Dugundji, Topology, Allyn & Bacon, Boston, Mass., 1966.
H. Freudenthal, Die Topologie der Lieschen Gruppen als algebraisches Phänomen. I, Ann. of Math. (2) 42 (1941), 1051–1074.
H. Freudenthal, H. de Vries, Linear Lie Groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York, 1969.
P. Gartside, B. Pejić, Uniqueness of Polish group topology, Topology Appl. 155 (2008), no. 9, 992–999.
В. М. Глышков, Строение локально бикомпактных групп и пятая проблема Гильберта, УМН 12 (1957), вьш. 2(74), 3–41. Engl. transl.: V. M. Gluškov, The structure of locally compact groups and Hilbert’s fifth problem, Amer. Math. Soc. Transl. (2) 15 (1960), 55–93.
M. Goto, On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc. 20 (1969), 157–162.
H. Gündoğan, The component group of the automorphism group of a simple Lie algebra and the splitting of the corresponding short exact sequence, J. Lie Theory 20 (2010), no. 4, 709–737.
S. Hernández, K. H. Hofmann, S. A. Morris, Nonmeasurable subgroups of compact groups, J. Group Theory 19 (2016), no. 1, 179–189.
E. Hewitt, K. A. Ross, Abstract Harmonic Analysis, Vol. I, 2nd ed., Springer, Berlin, 1979.
J. Hilgert, K. H. Hofmann, J. D. Lawson, Lie Groups, Convex Cones, and Semigroups, Oxford Mathematical Monographs, Oxford Univ. Press, New York, 1989.
J. Hilgert, K.-H. Neeb, Structure and Geometry of Lie Groups, Springer Monographs in Mathematics, Springer, New York, 2012.
K. H. Hofmann, L. Kramer, Transitive actions of locally compact groups on locally contractible spaces, J. Reine Angew. Math. 702 (2015), 227–243. Erratum, J. Reine Angew. Math. 702 (2015), 245–246.
K. H. Hofmann, S. A. Morris, Transitive actions of compact groups and topological dimension, J. Algebra 234 (2000), no. 2, 454–479.
K. H. Hofmann, S. A. Morris, Open mapping theorem for topological groups, Topology Proc. 31 (2007), no. 2, 533–551.
K. H. Hofmann, S. A. Morris, The structure of almost connected pro-Lie groups, J. of Lie Theory 21 (2011), 341–383.
K. H. Hofmann, S. A. Morris, The Structure of Compact Groups, 3rd ed., De Gruyter Studies in Mathematics, Vol. 25, de Gruyter, Berlin, 2013.
K. H. Hofmann, S. A. Morris, Pro-Lie groups: A survey with open problems, Axioms 4 (2015), 294–312.
K. Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558.
R. R. Kallman, The topology of compact simple Lie groups is essentially unique, Advances in Math. 12 (1974), 416–417.
R. R. Kallman, A uniqueness result for a class of compact connected groups, in: Conference in Modern Analysis and Probability (New Haven, Conn., 1982), Contemp. Math., Vol. 26, Amer. Math. Soc., Providence, RI, pp. 207–212.
A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, Vol. 156, Springer, New York, 1995.
J. O. Kiltinen, On the number of field topologies on an infinite field, Proc. Amer. Math. Soc. 40 (1973), 30–36.
L. Kramer, The topology of a semisimple Lie group is essentially unique, Adv. Math. 228 (2011), no. 5, 2623–2633.
K. Kuratowski, Topology, Vol. I, New ed., Academic Press, New York, 1966.
M. W. Liebeck et al., Commutators in finite quasisimple groups, Bull. Lond. Math. Soc. 43 (2011), no. 6, 1079–1092.
D. Montgomery, L. Zippin, Topological Transformation Groups, Interscience Publishers, New York, 1955.
S. Murakami, On the automorphisms of a real semi-simple Lie algebra, J. Math. Soc. Japan 4 (1952), 103–133.
N. Nikolov, D. Segal, On finitely generated profinite groups. I. Strong completeness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171–238.
J. C. Oxtoby, Measure and Category, 2nd ed., Graduate Texts in Mathematics, Vol. 2, Springer, New York, 1980.
B. J. Pettis, On continuity and openness of homomorphisms in topological groups, Ann. of Math. (2) 52 (1950), 293–308.
D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Graduate Texts in Mathematics, Vol. 80, Springer, New York, 1996.
A. I. Shtern, Van der Waerden continuity theorem for semisimple Lie groups, Russ. J. Math. Phys. 13 (2006), no. 2, 210–223.
A. I. Shtern, Bounded structure and continuity for homomorphisms of perfect connected locally compact groups, Proc. Jangjeon Math. Soc. 15 (2012), no. 3, 235–240.
T. E. Stewart, Uniqueness of the topology in certain compact groups, Trans. Amer. Math. Soc. 97 (1960), 487–494.
M. Stroppel, Locally Compact Groups, EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zürich, 2006.
J. Tits, Homorphismes “abstraits” de groupes de Lie, in: Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), Academic Press, London, 1974, pp. 479–499.
B. L. van derWaerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933), no. 1, 780–786.
F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Vol. 94, Springer, New York, 1983.
G. Warner, Harmonic Analysis on Semi-simple Lie Groups. I, Springer, New York, 1972.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
L. Kramer Partially supported by SFB 878.
Rights and permissions
About this article
Cite this article
BRAUN, O., HOFMANN, K.H. & KRAMER, L. AUTOMATIC CONTINUITY OF ABSTRACT HOMOMORPHISMS BETWEEN LOCALLY COMPACT AND POLISH GROUPS. Transformation Groups 25, 1–32 (2020). https://doi.org/10.1007/s00031-019-09537-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-019-09537-4