We investigate the problem of type isomorphisms in the presence of
higher-order references. We first introduce a finitary programming language
with sum types and higher-order references, for which we build a fully abstract
games model following the work of Abramsky, Honda and McCusker. Solving an open
problem by Laurent, we show that two finitely branching arenas are isomorphic
if and only if they are geometrically the same, up to renaming of moves
(Laurent's forest isomorphism). We deduce from this an equational theory
characterizing isomorphisms of types in our language. We show however that
Laurent's conjecture does not hold on infinitely branching arenas, yielding new
non-trivial type isomorphisms in a variant of our language with natural
numbers
