Growth of groups with incompressible elements, I

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.