Existence of group nonexpansive retractions and ergodic theorems in topological groups

Suppose that $G$ is a topological group and $C$ a compact subset of $G$. In this paper we define group nonexpansive mappings and then we consider $\sc = {T_{i} : i \in I }$asafamilyofthegroupnonexpansivemappingson$C$. Also we study the existence of group nonexpansive retractions $P_i$ from $C$ onto $\text{Fix}(\text{sc})$ such that $P_i{T}_i=T_i{P}_i=P_i$.