Triangle group representations and constructions of regular maps (Q2766398)

For integers \(m,n\geq 3\), the triangle group has the presentation NEWLINE\[NEWLINET(2,m,n)=\langle y,z\mid y^m=z^n=(yz)^2=1\rangle.NEWLINE\]NEWLINE The pair \(\{m,n\}\) is hyperbolic if \(1/m+1/n<1/2\). ``The main aim of this article is to \dots show that linear representations of triangle groups play a central role in the algebraic theory of regular maps.'' NEWLINENEWLINENEWLINESections 1 and 2 provide rich historical background and lucid motivation for the sequel. In \S 3 a group \(G\) is defined to be residually finite if for any finite \(M\) such that \(1_G\notin M\subset G\), there exists \(N\triangleleft G\) of finite index disjoint from \(M\). It is shown that residual finiteness of \(T(2,m,n)\) when \(1/m+1/n\leq 1/2\) is equivalent to the existence of certain finite regular maps on orientable surfaces of positive genus. In \S 5, ``one of the main goals \dots{} is to derive a method for converting faithful representations in linear groups over polynomial rings \dots{} into \(r\)-locally faithful representations in finite groups.'' This is then applied to the triangle groups. NEWLINENEWLINENEWLINEA map on a compact surface \(\mathcal{S}\) of positive genus is said to have planar width larger than \(r\) if every non-contractible simple closed curve on \(\mathcal{S}\) intersects the underlying graph of \(M\) at more than \(r\) points. Let \({\mathcal G}_r(m,n)\) denote the collection of underlying graphs of \(m\)-covalent, \(n\)-valent maps having planar width larger than \(r\). The following two results appear in \S 6. NEWLINENEWLINENEWLINETheorem 5. For every \(r\) and every hyperbolic pair \(\{m,n\}\), there exists a map in \({\mathcal G}_r(m,n)\) with at most \(C^r\) edges for some \(C<2^{72(m+n)(mn)^3}\). NEWLINENEWLINENEWLINETheorem 6. Let \(\{m,n\}\) be hyperbolic and \(r\geq m\). If \(m\) has no prime divisor \(\leq 2n\), then every graph in \({\mathcal G}_r(m,n)\) is an arc-transitive non-Cayley graph. NEWLINENEWLINENEWLINEDiscussion of improving the bound for \(C\) appears in \S 7.