On some groups related with Fox-Artin wild arcs (Q2799081)

In [Proc. Sympos. Pure Math. 1, 64--87 (1959; Zbl 0097.01406)], \textit{H. S. M. Coxeter} affirms that there are at least four ways where the binary polyhedral groups arise. In particular he says that these groups arise topologically as the fundamental groups of certain three manifolds discovered by Seifert and Threlfall. In the beautiful and interesting paper under review, the author generalizes certain symmetric group presentations found by Coxeter. The goal of the paper is to show that two different topics are related, these generalized groups and the ideal compactification of the sets obtained by lifting knots in 3-manifolds to their universal covering space. NEWLINENEWLINEThese generalized groups are obtained as follows. Consider a two bridge link \(K_{p/q}\) in \(S^3\), so each of the components is trivial, and performing \(0\)-surgery in one component we obtain a knot \(\mathbf K_{p/q}\) in \(S^1\times S^2\). Let \(\tilde K_{p/q}\) be the preimage of \(\mathbf K_{p/q}\) in the universal cover \({\mathbb {R}}^1\times S^2\). Then the ideal compactification of this cover is \((S^3, K'_{p/q})\) with \(K'_{p/q} = \tilde K_{p/q} \cup \epsilon({\mathbb{R}}^1 \times S^2)\), where \(\epsilon({\mathbb{R}}^1 \times S^2) \) is the end space of \(({\mathbb{R}}^1 \times S^2)\). Let \(l\) be the linking number between the components of \(K_{p/q}\); if \(| l | \not=0\), \(K'_{p/q}\) consists of \(| l |\) arcs in \(S^3\) sharing their endpoints, \(0\) and \(\infty\), which is the wild set of \(K'_{p/q}\). Denote \(\pi _1 (S^3 - K'_{p/q})\) by \(G(p/q)\), this group has a presentation with generators \(B_n\) for all \(n\in {\mathbb{Z}}\) and relations NEWLINE\[NEWLINEB^{e_1}_n B^{e_3}_{n+e_2} B^{e_5}_{n+e_2+e_4}\dots B^{e_{p-3}}_ {n+e_2+\dots+e_{p-4}}B^{e_{p-1}}_{n+e_2+\dots+ e_{p-2}}NEWLINE\]NEWLINE Denote by \(G(p/q)_r\) the group presented by the generators \(B_0, B_1,\dots, B_{r-1}\) and the \(r\) relations NEWLINE\[NEWLINE B^{e_1}_n B^{e_3}_{n+e_2} B^{e_5}_{n+e_2+e_4}\dots B^{e_{p-3}}_ {n+e_2+\dots +e_{p-4}}B^{e_{p-1}}_{n+e_2+\dots+ e_{p-2}},NEWLINE\]NEWLINE NEWLINE\[NEWLINEn\in \{0,1,\dots,r-1\},NEWLINE\]NEWLINE where the subindices are taken mod \(r\), and where \(e_i\), \(1\leq i \leq p-1\), is the sign (plus or minus) of \(iq\) reduced mod \(2p\) in the interval \((-p,p)\). This last group is an epimorph of \(G(p/q)\) for every \(r\geq 1\) and the epimorphism sends the generator \(B_n\) to \( B_{n\text{ mod }r}\). A geometrical interpretation of these groups is given as follows. Let \(U\) be a component of a 2-bridge link \(K_{p/q}\) and consider the \(r\)-fold covering \(p_r : S^3\rightarrow S^3\) branched over \(U\). Let \(K_{p/q}^r\) be \(p_r^{-1} (K_{p/q})\). Performing 0-surgery in the unknot \(U\) we obtain \(S^1\times S^2\) and a knot or link \(\mathbf {K}^r_{p/q}\). Let \(a\) be a meridian of \(U\) and let \((S^1\times S^2 - \mathbf K_{p/q}^r)\cup_a Q^2\) be the result of attaching a 2-cell along \(a\). Then \(G(p/q)_r\) is isomorphic to \(\pi _1 ((S^1\times S^2 -\mathbf K_{p/q}^r)\cup_a Q^2)\). These groups also have a four dimensional interpretation, as the fundamental group of \(B^4\) minus the mapping cylinder of the natural projection of \(S^1\times S^2 \) to \(S^1\) restricted to \(\mathbf K_{p/q}^r\). Several examples are worked, as the Fox-Artin wild arc [\textit{R. H. Fox} and \textit{E. Artin}, Ann. Math. (2) 49, 979--990 (1948; Zbl 0033.13602)], which is the arc \(K'_{14/3}\), and the group \(G(14/3)_2\) is the binary icosahedral group.