  We define the class of groups of bounded type with finite cycles from tile
inflations. These tile inflations also determine some automata describing the
groups. In the case when the automata are finite-state, we show that if the set
of incompressible elements of a group in this class is finite, then this group
has subexponential growth with a bounded power in the exponent. Then we discuss
some special cases with certain special structures of orbital graphs and give
explicit ways to find the upper bounds for the growth functions of these
groups.
