Isogeny graphs on superspecial abelian varieties: eigenvalues and connection to Bruhat-Tits buildings (Q6650483)

Let \(\mathcal{SS}_g(p)\) denote the set of isomorphiam classes of principally polarized superspeial abelian varieties \((A, \mathcal{L})\) of dimension \(g\) over \(\overline{\mathbb{F}}_p\). Fix a representative \((A_0, \mathcal{L}_0)\) of a class in \(\mathcal{SS}_g(p)\), then there exists an isogeny \(\phi_A:A_0\rightarrow A\) of \(\ell\)-power degree such that \(\mathrm{Ker}(\phi_A)\) is a maximal totally isotropic subspace of \(A[\ell^n]\) for some \(n\geq 0\). Let \(\mathcal{SS}_g(p,\ell, A_0, \mathcal{L}_0)\) denote the set of equivalence classes of triples \((A,\mathcal{L},\phi_A)\), where two triples are defined to be equivalent if there exists an isomorphism between them compatible with the isogeny. \N\NThe main object of this paper, the isogeny graph \(\mathcal{G}_g^{SS}(\ell, p)\), is a directed graph such that the set of vertices is \(\mathcal{SS}_g(p,\ell, A_0, \mathcal{L}_0)\), and the set of edges connecting two vertices \(v_1, v_2\) is the equivalence classes of \(\ell^g\)-isogenies between corresponding polarized abelian varieties which commute with isogenies defining \(v_1, v_2\). The graph \(\mathcal{G}_g^{SS}(\ell, p)\) is regular since it has \(\prod_{k=1}^g(\ell^k+1)\)-outgoing edges from each vertex. For a regular directed graph \(\mathcal{G}\) of degree \(d\), the normalized Laplacian \(\Delta\) is defined to be \(E-(1/d)M\) where \(M\) is the adjacency matrix of \(\mathcal{G}\). \N\NThe first main result of the present paper shows that there exists a positive constant \(c_{g,\ell}\) such that the second smallest eigenvalue of the normalized Laplacian is greater than or equal to \(c_{g,\ell}\) for any \(p\neq \ell\). The second result shows that there exist natural isomorphisms between three graphs \(\mathcal{G}_g^{SS}(\ell, p)\), the big isogeny graph \(Gr_g(\ell, p)\) introduced in [\textit{B. W. Jordan} and \textit{Y. Zaytman}, ``Isogeny graphs of superspecial abelian varieties and Brandt matrices'', Preprint, \url{arXiv:2005.09031}], and a finite quotient graph \(\Gamma\backslash\mathcal{S}_g\) of a 1-complex \(\mathcal{S}_g\) defined in terms of the Bruhat-Tits building for \(PGSp_g(\mathbb{Q}_{\ell})\). This implies a rapid mixing property of natural random walks on these families of graphs.