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Cyclically presented groups with length four positive relators

For cyclically presented groups $G = G_n(w)$ with positive length four relators $w = x_0x_jx_kx_l$ in the free group with basis $x_0, x_1, \ldots, x_{n-1}$, we classify finiteness and, modulo two unresolved cases, we classify asphericity for the underlying presentations. We show that the fixed point subgroup of the shift $x_i \mapsto x_{i+1}$ is always finite and we relate finiteness of $G$ and asphericity to the dynamics of the shift action by the cyclic group of order $n$ on the nonidentity elements of $G$.