  Extended geometry provides a unified framework for double geometry,
exceptional geometry, etc., i.e., for the geometrisations of the string theory
and M-theory dualities. In this talk, we will explain the structure of gauge
transformations (generalised diffeomorphisms) in these models. They are
generically infinitely reducible, and arise as derived brackets from an
underlying Borcherds superalgebra or tensor hierarchy algebra. The infinite
reducibility gives rise to an $L_\infty$ structure, the brackets of which have
universal expressions in terms of the underlying superalgebra.
