3 problems

Let $G$ be a group with a symmetric generating set $S$, and suppose that the geodesic growth of $(G,S)$ is polynomial. Polynomial-growth conjecture. Then $G$ is virtually $2$-step…

Let $\mathcal{v\!H}$ be the virtually nilpotent group considered above, and let $S$ be its generating set. The geodesic growth rate with respect to $S$ is the number of geodesics o…

Let $T$ and $T'$ be trees. Assume that they are simultaneously co-spectral, their complements are co-spectral, and their line graphs are co-spectral. Geodesic-growth conjecture. Th…