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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. [ABSTRACT FROM AUTHOR]

Abstract The commutator subgroup of a free group is the subgroup consisting of elements contained in the kernel of every homomorphism from the free group to the integers. We give a similar characterization of the second derived subgroup of a free group. Specifically, we show that the second derived subgroup of a free group is equal to the intersection of the kernels of all homomorphisms into a solvable deficiency 1 group. [ABSTRACT FROM AUTHOR]

We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group. [ABSTRACT FROM AUTHOR]

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor lim ← 1 , which is, however, notoriously difficult to compute when the fundamental group, π 1 (X) , is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit X ˜ of a sequence of covering spaces in terms of the functors lim ← and lim ← 1 applied to sequences of countable groups arising from polyhedral approximations of X. A consequence of our computation is that path-connectedness of a lifting space, X ˜ , implies that π 1 (X ˜) supplements π 1 (X) in π ˇ 1 (X) where π ˇ 1 (X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that G ⋅ Ker Z (F ˆ) = F ˆ ≠ G ⋅ Ker B (1 , n) (F ˆ) , where F ˆ is the canonical inverse limit of finite rank free groups, G is the fundamental group of the Hawaiian Earring, B (1 , n) is the Baumslag-Solitar group, and Ker A (F ˆ) is the intersection of kernels of homomorphisms from F ˆ to A. [ABSTRACT FROM AUTHOR]