Growth of groups with incompressible elements, I

We define the class of groups of bounded type from tile inflations. These tile inflations also determine some automata describing the groups. In the case when the automata are stationary, we show that if the set of incompressible elements of a group in this class is finite, then this group has subexponential growth whose growth function is bounded above by functions of the form \exp(\textit{CR}^{\alpha}) for R>0 and \alpha\in(0,1) . Then we describe some examples with certain special structures of orbital graphs and give explicit ways to find the upper bounds for the growth functions of these groups.