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Let $B$ be a real Banach space and let $G$ be a finitely generated group acting on $B$. Suppose that for every $g \in G$, the spectral radius of $g$ is $1$. Suppose moreover that there exists a $G$-stable closed (for the weak topology) convex cone $C$ in $B$ which doesn't contain any linear subspace. Then we know that for every $g \in G$, there exists $x \in C - \{0\}$ such that $g(x) = x$ (see for instance this paper). Can we find $x \in C - \{0\}$ such that $g(x) = x$ for all $g \in G$?

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