  In recent work it was shown how recursive factorisation of certain QRT maps
leads to Somos-4 and Somos-5 recurrences with periodic coefficients, and to a
fifth-order recurrence with the Laurent property. Here we recursively factorise
the 12-parameter symmetric QRT map, given by a second-order recurrence, to
obtain a system of three coupled recurrences which possesses the Laurent
property. As degenerate special cases, we first derive systems of two coupled
recurrences corresponding to the 5-parameter multiplicative and additive
symmetric QRT maps. In all cases, the Laurent property is established using a
generalisation of a result due to Hickerson, and exact formulae for degree
growth are found from ultradiscrete (tropical) analogues of the recurrences.
For the general 18-parameter QRT map it is shown that the components of the
iterates can be written as a ratio of quantities that satisfy the same Somos-7
recurrence.
