A Whitehead algorithm for surface groups
A problem about a group and its automorphisms: Given finite sets \(\{a_1,\dots,a_n\}\) and \(\{b_1,\dots,b_n\}\) of elements of a group \(G\), is there an automorphism of \(G\) taking \(a_i\) to \(b_i\) for all \(i\)? If so, find one. In 1936, J.~H.~C. Whitehead found an elegant algorithm which solves this problem for a finitely generated free group. In 1984, Collins and Zieschang extended Whitehead's methods to free products of finitely many freely indecomposable groups, assuming that the above problem, which will be called the Whitehead problem, can be solved in each factor. In this paper, the authors give an algorithm to solve the Whitehead problem for the fundamental group of a closed surface \(\Sigma\) and a surface with boundary. If one represents the elements \(a_i\) and \(b_i\) in \(\pi_1(\Sigma,p)\) by loops, the problem then translates into the following problem about closed curves on a surface: Is there a diffeomorphism of \(\Sigma\) fixing \(p\) which takes each \(\alpha_i\) to a loop homotopic to \(\beta_i\)? If so, find one. The solution to this geometric problem relies in part on work of Hass and Scott on straightening curves on surfaces.
scientific article; zbMATH DE number 1526239