# A note on automatic continuity

@article{Conner2018ANO,
title={A note on automatic continuity},
author={Gregory R. Conner and Samuel M. Corson},
journal={Proceedings of the American Mathematical Society},
year={2018}
}
• Published 13 October 2017
• Mathematics
• Proceedings of the American Mathematical Society
We present new results regarding automatic continuity, unifying some diagonalization concepts that have been developed over the years. For example, any homomorphism from a completely metrizable topological group to Thompson's group $F$ has open kernel. A similar claim holds when $F$ is replaced with a Baumslag-Solitar group or a torsion-free word hyperbolic group.
17 Citations
INFINITE-TYPE SURFACES AND MAPPING CLASS GROUPS: OPEN PROBLEMS
• Mathematics
• 2021
These are notes on the open problem session run by Priyam Patel and Nicholas Vlamis for the infinite-type surfaces group at the 2021 Nearly Carbon Neutral Geometric Topology conference organized by
Inverse limit slender groups
• Mathematics
• 2021
Classically, an abelian group G is said to be slender if every homomorphism from the countable product Z to G factors through the projection to some finite product Z. Various authors have proposed
Automatic continuity for groups whose torsion subgroups are small
• Mathematics
Journal of Group Theory
• 2022
Abstract We prove that a group homomorphism φ : L → G \varphi\colon L\to G from a locally compact Hausdorff group 𝐿 into a discrete group 𝐺 either is continuous, or there exists a normal open
Mapping class groups with the Rokhlin property
• Mathematics
• 2021
We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager
Generating the inverse limit of free groups
• Mathematics
Journal of Algebra
• 2021
Automatic continuity of ℵ1-free groups
We prove that groups for which every countable subgroup is free (ℵ1-free groups) are n-slender, cm-slender, and lcH-slender. In particular, every homomorphism from a completely metrizable group to an
Abstract homomorphisms from some topological groups to acylindrically hyperbolic groups
• Mathematics
Mathematische Annalen
• 2021
We describe homomorphisms $\varphi:H\rightarrow G$ for which the codomain is acylindrically hyperbolic and the domain is a topological group which is either completely metrizable or locally countably

## References

SHOWING 1-10 OF 14 REFERENCES
On Endomorphisms of torsion-Free hyperbolic Groups
• Mathematics
Int. J. Algebra Comput.
• 2011
A mixed (con jugate/non-conjugate) version of the classical Whitehead problem for tuples is solvable in torsion-free hyperbolic groups.
Torsion-free word hyperbolic groups are noncommutatively slender
This result demonstrates that random finitely presented groups in the few- relator sense of Gromov are noncommutatively slender.
Atomic Properties of the Hawaiian Earring Group for HNN Extensions
In 2011, while investigating fundamental groups of wild spaces, K.Eda [7] showed that the fundamental group of the Hawaiian earring (the Hawaiian earring group, in short) has the property that for
Combinatorial Group Theory
• Mathematics
• 1977
Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7.