부산대학교 도서관

Let $F$ denote a local non-Archimedean field and let ${\rm PGL}(n,F)$ denote a projective general linear group over $F$. Then the Bruhat-Tits building $\scr{B}_{n}$ associated to ${\rm PGL}(n,F)$ is a contractable $(n-1)$-dimensional simplicial complex. One obtains finite complexes $X_{\Gamma}$ by taking quotients of $\scr{B}_{n}$ by torsion-free discrete cocompact subgroups $\Gamma$ of ${\rm PGL}(n,F)$. \par It is of particular interest to study an extension of zeta function, counting closed geodesic tailless cycles in $X_{\Gamma}$, to complexes. In the paper under review, the authors study the Bruhat-Tits building $\scr{B}_{3}$ and prove that the zeta function of associated finite complex $X_{\Gamma}$ is a rational function with two different closed form expressions. Further, it satisfies the Riemann hypothesis if and only if $X_{\Gamma}$ is a Ramanujan complex.