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The connected covering spaces of a connected and locally path-connected topological space $X$ can be classified by the conjugacy classes of those subgroups of $pi_1(X,x)$ which contain an open normal subgroup of $pi_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring $mathds{H}$ with corresponding open subgroup of $pi_1(mathds{H})$ which does not contain {em any} nontrivial normal subgroup of $pi_1(mathds{H})$.
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a sur
We prove that some classes of triangle-free Artin groups act properly on locally finite, finite-dimensional CAT(0) cube complexes. In particular, this provides the first examples of Artin groups that are properly cubulated but cannot be cocompactly c
We classify those compact 3-manifolds with incompressible toral boundary whose fundamental groups are residually free. For example, if such a manifold $M$ is prime and orientable and the fundamental group of $M$ is non-trivial then $M cong Sigmatimes S^1$, where $Sigma$ is a surface.
Let $t_1,ldots,t_n$ be $ell$-group terms in the variables $X_1,ldots,X_m$. Let $hat t_1,ldots,hat t_n$ be their associated piecewise homogeneous linear functions. Let $G $ be the $ell$-group generated by $hat t_1, ldots,hat t_n$ in the free $m$-gener
Let $$1 to H to G to Q to 1$$ be an exact sequence where $H= pi_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide sufficient con