  A connected linear algebraic group G is called a Cayley group if the Lie
algebra of G endowed with the adjoint G-action and the group variety of G
endowed with the conjugation G-action are birationally G-isomorphic. In
particular, the classical Cayley map, X \mapsto (I_n-X)/(I_n+X), between the
special orthogonal group SO_n and its Lie algebra so_n, shows that SO_n is a
Cayley group. In an earlier paper (see math.AG/0409004) we classified the
simple Cayley groups defined over an algebraically closed field of
characteristic zero. Here we consider a new numerical invariant of G, the
Cayley degree, which "measures" how far G is from being Cayley, and prove upper
bounds on Cayley degrees of some groups.
