Let $G=N\rtimes H$ be a finite Frobenius group with $|N|=q$ and $|H|=q-1$. We classify all Ramanujan Cayley graphs of $G$ whose connection sets are normal, in the sense of being unions of conjugacy classes. The group-theoretic input is a simple blow-up phenomenon: every such Cayley graph is either $Y[\overline{K_q}]$ or $Y[K_q]$ for a connected regular Cayley graph $Y$ on the complement $H$. We first prove a graph-theoretic result classifying all Ramanujan graphs of these two forms when $Y$ is an arbitrary connected regular graph on $q-1$ vertices. The proof combines the classical characterization of regular graphs with least eigenvalue greater than $-2$ with a second-moment identity in the bipartite case. Translating the resulting five graph types back to $G$ yields a complete classification for all ratio-one Frobenius groups, and in particular for $\operatorname{AGL}(1,q)$ over every finite field.
Let $\mathscr{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_n(F)$, where $F$ is a non-Archimedean local field. We introduce geometric $k$-geodesics in $\mathscr{B}$ by means of CAT(0) convexity and combinatorial $k$-geodesics by a local successor relation on pointed $k$-facets. We prove that the two notions coincide. This allows us to use the local combinatorial definition on quotients $\Gamma\backslash\mathscr{B}$, without referring to the universal covering. When $\Gamma$ is discrete, torsion-free, cocompact, and type-preserving, the primitive closed $k$-geodesics define zeta functions $Z_k$ and their $\epsilon$-twisted variants $Z_k^\epsilon$. Our main result identifies an alternating product of these zeta functions with the unramified $L$-function of $L^2(\Gamma\backslash \mathrm{PGL}_n(F))$: $(1-u^n)^{\chi(\Gamma\backslash\mathscr{B})}L(\Gamma,q^{(n-1)/2}u)=\prod_{k=1}^{n-1} Z_k^\epsilon(\Gamma\backslash\mathscr{B},u)^{(-1)^{k+1}}$. This gives a uniform Ihara-type identity for all $\mathrm{PGL}_n$. We also extend the construction and the identity to $\mathrm{PGL}_n(D)$, where $D$ is a central division algebra over $F$; in that setting the residue parameter is $Q=|\mathcal{O}_D/\mathfrak{p}_D|$.
Ming-Hsuan Kang, Jiu-Kang Yu· 0 citations
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