In this short paper the author summarises some of the interactions between group actions in hyperbolic geometry and low-dimensional topology, including theorems of Hurwitz and Siegel and the Gauss-Bonnet index theorem for Fuchsian groups. He also re-visits a theorem of A. M. Macbeath [Proc. Glasg. Math. Assoc. 5, 90–96 (1961; Zbl 0134.16603)], which is now known as the ‘Macbeath trick’, and can be stated as follows: If there exists a Riemann surface \(S\) of genus \(g > 1\) admitting a group of \(h(g-1)\) automorphisms, where \(h\) is a rational number with denominator dividing \(g-1\), then for infinitely many values of the integer \(k\) there exists a \(k\)-sheeted covering surface \(S_k\) of genus \(g_k = k^{2g}(g-1)+1\) admitting a group of \(h(g_k-1)\) automorphisms. The proof involves an infinite sequence of characteristic subgroups of the fundamental group \(K\) of the surface \(S\) (all containing \(K' = [K,K]\)), and Macbeath used this to prove that the Hurwitz bound of \(84(g-1)\) on the number of conformal automorphisms of \(S\) is attained for infinitely many values of \(g\). Finally, the author explains a potential extension of Macbeath’s theorem to hyperbolic 3-manifolds.
For the entire collection see [Zbl 1348.05004].