Universität Kiel

Gravitational instantons are complete 4-dimensional hyperkähler manifolds with square-integrable curvature tensor. I will address the question whether all gravitational instantons (of type ALG) can be obtained as Hitchin moduli spaces. In particular, I will explain how to compute the (hyperkähler) Torelli map for (weakly) parabolic Higgs bundles on the 4-punctured sphere. This is based on recent joint work with Fredrickson, Mazzeo and Swoboda.

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).