The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field $k$) modulo the relation that for a closed immersion $Y\to X$ there is a relation that $[X] = [Y] + [X \setminus Y]$. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field $k$ is finite for a variety $X$ we can consider the "almost-finite" set $X(\bar k)$, which represents the local zeta function of $X$. In this talk we will discuss how to detect interesting elements in $K_1(\text{Var})$ (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on $X$.

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