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For an arbitrary sequence    (   G  s   )    of subgroups of finite index in the generalized Thompson group    F   n  ,  ∞    , it is shown that the supremum of the minimal number of generators of    G  s    is finite and the supremum of the absolute value of the deficiency of    G  s    is finite. In particular, the deficiency gradient of    F   n  ,  ∞with respect to    (   G  s   )    is 0 provided    [  G  :   G  s   ]    tends to infinity. A higher dimensional analogue is considered for    n  =  2   .

Rank and deficiency gradients of generalized Thompson groups of type F