Cyclically presented groups with length four positive relators

Abstract

For cyclically presented groups $G=G_n(w)$ with positive length four relators $w=x_0{x}_j{x}_k{x}_l$ in the free group with basis $x_0,x_1,\dots ,x_{n-1}$, 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 $G$.

arXiv comment33pp, v3 fixes one typo from v2 (def of types I*, U* on page 4)