Relation Between Algebraic Properties and Dynamical Properties Of Group Actions

Let : G X ! X be a group action on a compact metric space (X; d). In this paper, we introduce finite expansive group action, elastic group action and show that nilpotent group actions on S 1, have no finite expansive. Also we show that if H is finite index subgroup of G, then : G X ! X is finite expansive if and only if jH : H X ! X is finite expansive. We show that nilpotent group actions have no elastic on a closed interval but there is solvable group action on a closed interval, that is elastic.