The fundamental group of the Menger universal curve is uncountable and not free, although all of its finitely generated subgroups are free. It contains an isomorphic copy of the fundamental group of every one-dimensional separable metric space and an
isomorphic copy of the fundamental group of every planar Peano continuum. We give an explicit and systematic combinatorial description of the fundamental group of the Menger universal curve and its generalized Cayley graph in terms of word sequences. The word calculus, which requires only two letters and their inverses, is based on Pasynkovs partial topological product representation and can be expressed in terms of a variation on the classical puzzle known as the Towers of Hanoi.
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 nat
ural 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 $pi_1(mathds{H})$.
In their study of fundamental groups of one-dimensional path-connected compact metric spaces, Cannon and Conner have asked: Is there a tree-like object that might be considered the topological Cayley graph? We answer this question in the positive and
provide a combinatorial description of such an object.
