Biautomatic groups arose as groups explaining formal language-theoretic aspects of geodesics in word-hyperbolic groups.  Many classes of non-positively curved finitely generated groups, such as hyperbolic, virtually abelian, cocompactly cubulated, small cancellation and Coxeter groups, are known to be biautomatic.  On the other hand, there are some other classes, such as CAT(0) or hierarchically hyperbolic groups, for which the relationship to biautomaticity is more complicated.

In the first half of the talk, I will outline the notions of non-positive curvature appearing in group theory and their connection to biautomaticity.  In particular, I will overview recent results on the relationship between biautomaticity, hierarchical hyperbolicity and being CAT(0), as well as some constructions of non-biautomatic non-positively curved groups.

The goal of the second half of the talk is to construct a non-biautomatic hierarchically hyperbolic group, giving the first known example of such a group.  Our group acts geometrically on the cartesian product of a tree and the hyperbolic plane, and therefore satisfies many nice geometric properties.  The proof of non-biautomaticity will rely on the study of geodesic currents on a closed hyperbolic surface.  The talk is based on joint work with Sam Hughes.