A transcendental meromorphic function is called a Speiser
function if it has only
finitely many singular values (i.e., critical and asymptotic values). We
show how to
construct such functions with any given Hausdorff dimension of escaping
set by using
quasiconformal mappings. This strengthens an earlier result of
Bergweiler and Kotus. Our construction also shows that the escaping sets of
quasiconformally equivalent
meromorphic functions may be different. This work is joint with Magnus
Aspenberg.