We discuss the notion of growth for discrete quantum groups, with a number of general considerations, and with some explicit computations. Of particular interest is the quantum analogue of Gromov's estimate regarding polynomial growth: we formulate the precise question, and we verify it for the duals of classical Lie groups. (1) Deformation conjecture. This states that for any compact quantum group, there are 1 ≤ n < ∞ compact quantum groups of Kac type having the same (R + , dim) invariant.

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... "anti-deformation" inequality n ≥ 1 was checked in many situations, see [1] . As for the finiteness inequality n < ∞, this holds in the finite quantum group case, by a result of Stefan [18] .