Examining the Burnside Problem on Diff∞ ω (S2)

Abstract

Let $G$ be a group. $G$ is called \emph{periodic} if, for every element $g \in G$, there is a positive integer $n \in \N$ such that $g^n = \id_G$, and \emph{periodic of bounded exponent} if one such $n$ works for all $g \in G$. A question of interest in group theory asks whether $G$ being finitely generated and periodic (possibly periodic of bounded exponent) is enough to ensure that $G$ is finite. While this question is settled for arbitrary groups, it is still mostly open in the case where $G$ is a nonlinear transformation group acting on a manifold. We follow the proof from Hurtado et al. that, in the case of $\Diff_{\omega}^{\infty} (\Sp^2)$, every finitely generated periodic subgroup of bounded exponent is finite, providing additional context that was passed over in its original presentation. We then discuss the results from the appendix of that paper, before concluding with some remarks on how one might approach this type of problem.