The nonexistence of expansive polycyclic group actions on some Peano continua

Based on a fixed point theorem by Shi and Ye for amenable group actions on dendrites, we show that if X is a Peano continuum having the property that for any \(\epsilon >0\) $ϵ>0$ there are only finitely many circles in X with diameters \(\ge \epsilon \) $\ge ϵ$ , then there exists no expansive polycyclic group action on X. As a corollary, we obtain that local dendrites and Peano continua containing no \(\theta \) $\theta$ -curves admit no expansive polycyclic group actions.