Fast robbers on abelian Cayley graphs and digraphs

We study the fast-robber version of the Cops and Robbers game on finite strongly connected abelian Cayley digraphs, including undirected Cayley graphs as the symmetric case. For bounded out-degree $D$, we show that $c_{1,\mathrm{\infty }}\left(\mathrm{\Gamma }\right)=O_D\left(n^{1-\frac{1}{D}}\right)$; in the undirected case with $D\ge 2$, this improves to the optimal exponent $1-\frac{1}{\lfloor \frac{D}{2}\rfloor }$. We also establish the degree-independent bound $c_{1,\mathrm{\infty }}\left(\mathrm{\Gamma }\right)=O\text{\hspace{0.17em}⁣}\left(\frac{n{(\log \log n)}^2}{\log n}\right)$. These estimates follow from an optimized character-theoretic cyclic sweep over subgroup quotients.