We examine the equivariant birational linearisation problem for algebraic tori equipped with a finite group action. We also study bounds on the degree of linearisability, a measure of the obstruction for such an algebraic torus to be linearisable. We connect these problems to the question of determining when an algebraic group is (stably) Cayley - that is (stably) equivariantly birationally isomorphic to its Lie algebra.

We discuss joint work with Popov and Reichstein on the classification of the simple Algebraic groups which are Cayley and on determining bounds on the Cayley degree of an algebraic group, a measure of the obstruction for an algebraic group to be Cayley.

We also relate this to recent work with Borovoi, Kunyavskii and Reichstein extending the classification of stably Cayley simple groups from the algebraically closed characteristic zero case to arbitrary fields of characteristic zero. Lastly, we investigate the stable rationality of four-dimensional algebraic tori and the associated equivariant birational linearisation problem.