Some Results on Polish Groups

Gianluca Paolini,

Saharon Shelah

Abstrakt

We prove that no quantifier-free formula in the language of group theory can define the ℵ1-half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group. Finally, we prove some results on the structure of the group of automorphisms of a locally finite group: firstly, we prove that it is not the case that every group of automorphisms of a graph of power λ is the group of automorphism of a locally finite group of power λ; secondly, we conjecture that the group of automorphisms of a locally finite group of power λ has a locally finite subgroup of power λ, and reduce the problem to a problem on p-groups, thus settling the conjecture in the case λ = ℵ0.

AMS Subject Classification: 03E15, 20K30, 20B27.

Słowa kluczowe: Polish groups, automorphism groups, locally finite groups
References

[1] L. Fuchs, Infinite Abelian Groups – Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London 1970.

[2] L. Fuchs, Infinite Abelian Groups – Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973.

[3] A.S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.

[4] A.H. Mekler, Stability of Nilpotent Groups of Class 2 and Prime Exponent, J. Symbolic Logic 46:4 (1981), 781–788.

[5] A.S. Kechris, A.Nies, K. Tent, The Complexity of Topological Group Isomorphism, J. Symbolic Logic 83:3 (2018), 1190–1203.

[6] G. Paolini, S. Shelah, Groups Metrics for Graph Products of Cyclic Groups, Topology Appl. 232 (2017), 281–287.

[7] G. Paolini and S. Shelah, The Automorphism Group of Hall’s Universal Group, Proc. Amer. Math. Soc. 146 (2018), 1439–1445.

[8] Su Gao, Invariant Descriptive Set Theory, Pure and Applied Mathematics (Boca Raton), 293. CRC Press, Boca Raton, FL, 2009.

[9] S. Shelah, Beginning of Stability Theory for Polish Spaces, Israel J. Math. 214:2 (2016), 507–537.

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