Title: $\ell_1$ spreading models and the FPP for Cesàro mean nonexpansive maps

Abstract: Let $K$ be a nonempty set in a Banach space $X$. A mapping $T\colon K\to K$ is called {\it $\mathfrak{cm}$-nonexpansive} if for any sequence $(x_i)_{i=1}^n$ and $y$ in $K$, one has $\|(1/n) \sum_{i=1}^n Tx_i -Ty\|\leq \|(1/n)\sum_{i=1}^n x_i - y\|$. As a subsclass of nonexpansive maps, the FPP for such maps is well-established in a great variety of spaces. The main result of this paper is a fixed point result relating $\mathfrak{cm}$-nonexpansiveness, $\ell_1$ spreading models and Schauder bases with not-so-large basis constants. As a result, we deduce that every Banach space with weak Banach-Saks property has the fixed point property for $\mathfrak{cm}$-nonexpansive maps.