${\mathbb{Z}}_p$-folded surfaces and the ${\mathbb{Z}}_p$-Thurston norm

${\mathbb{Z}}_p$-folded surfaces are a class of oriented folded surfaces with a fundamental $H_2(\mathbb{Z}\mathrm{/}p\mathbb{Z})$ class. In this paper we study the ${\mathbb{Z}}_p$-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the ${\mathbb{Z}}_p$-Thurston norm, which generalises the ${\mathbb{Z}}_2$-Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev’s definition of $\theta$. We define canonical ${\mathbb{Z}}_p$-folded surfaces, which connect the theory of ${\mathbb{Z}}_p$-folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the ${\mathbb{Z}}_p$-Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty ${\mathbb{Z}}_3$-folded surface embedded in $L(3n,1)$ is $2-n$ for $n\in {\mathbb{Z}}^{>0}$, determining the ${\mathbb{Z}}_3$-Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space $L(3n,1)$ must have at least $n$ tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of $L(3n,1)$ using more sophisticated techniques.