On Borel Reducibility in Generalised Baire Space

In this paper we study the Borel reducibility of Borel equivalence relations, including some orbit equivalence relations, on the generalised Baire space ${\kappa }^{\kappa }$ for an uncountable $\kappa$ with the property ${\kappa }^{<\kappa }=\kappa$. The theory looks quite different from its classical counterpart where $\kappa =\omega$, although some basic theorems do generalise.