Multi-EGS groups: exponent of congruence quotients

Given a multi-EGS group K acting on the p-adic tree, where p is any prime number, we compute the exponent of the congruence quotient \(K_n= K/ {{\,\textrm{St}\,}}_K(n)\) $K_n=K/{\text{St}}_K(n)$ for all \(n\ge 1\) $n\ge 1$ . The formula that we obtain for \(\exp (K_n)\) $exp(K_n)$ only depends on p, n and the periodicity of K.