[Paper Review] Explicit Constructions of Ramanujan Complexes of Type $ ilde{A}_d$

Alexander Lubotzky, Beth Samuels+1 more|ArXiv.org|Jun 10, 2004

Graph theory and applications10 references3 citations

TL;DR

This paper presents explicit constructions of Ramanujan complexes of type $\tilde{A}_d$ for $d \geq 2$ over local fields of positive characteristic, using congruence quotients of the Cartwright-Steger lattice $\Gamma$ in $\mathrm{PGL}_d(F)$, where $F = \mathbb{F}_q((y))$. The key result is an explicit set of generators for $\mathrm{PGL}_d(\mathbb{F}_{q^e})$ such that the resulting Cayley complex is a Ramanujan complex covered by the Bruhat-Tits building.

ABSTRACT

In this paper we present for every $d \geq 2$ and every local field $F$ of positive characteristic, explicit constructions of Ramanujan complexes which are quotients of the Bruhat-Tits building $\B_d(F)$ associated with $\operatorname{PGL}_d(F)$.

Motivation & Objective

• To provide explicit, constructive realizations of Ramanujan complexes of type $\tilde{A}_d$ for $d \geq 2$ over local fields of positive characteristic.
• To extend the existence results of Ramanujan complexes to explicit constructions using finite quotients of the Cartwright-Steger lattice $\Gamma$.
• To realize the $1$-skeleton of the finite quotients as Cayley graphs of finite groups with explicit generators.
• To establish that these complexes are Ramanujan by verifying spectral properties via the global Jacquet-Langlands correspondence for function fields.
• To provide an algorithmic framework for constructing such complexes over finite local rings and finite fields, including a detailed example.

Proposed method

• Use the Cartwright-Steger lattice $\Gamma$, a discrete subgroup of $\mathrm{PGL}_d(F)$ acting simply transitively on the vertices of the Bruhat-Tits building $\mathcal{B}_d(F)$, as the foundational group.
• Construct finite quotients of $\Gamma$ via congruence subgroups $\Gamma(I)$, where $I = \langle p^s \rangle$ for a prime $p \in \mathbb{F}_q[y]$ and $s \geq 1$, leading to finite quotients $\Gamma(I) \backslash \mathcal{B}_d(F)$.
• Embed $\Gamma$ explicitly into $\mathrm{PGL}_d(L)$ for finite local rings $L = R/I$, using an explicit $d \times d$ matrix representation over $\mathbb{F}_q[x]$, enabling algorithmic construction.
• Define the Cayley complex as the simplicial complex whose $1$-skeleton is the Cayley graph $\mathrm{Cay}(G; S)$, with $G = \mathrm{PGL}_d(\mathbb{F}_{q^e})$ and $S$ a set of generators corresponding to subspaces of $\mathbb{F}_q^d$ of dimensions $1$ to $d-1$, with $|S| = \sum_{k=1}^{d-1} \genfrac{[}{]}{0pt}{2}{d}{k}_q$.
• Prove that the resulting complexes are Ramanujan by appealing to the spectral theory of $\mathrm{PGL}_d(F)$-actions and the global Jacquet-Langlands correspondence for function fields, as established in [LSV].
• Provide a detailed algorithm and example in Section 10 for constructing the generators as $d \times d$ matrices over $\mathbb{F}_q[x]$, with explicit generators listed in Figure 9.

Experimental results

Research questions

• RQ1 Can Ramanujan complexes of type $\tilde{A}_d$ be explicitly constructed for $d \geq 2$ over local fields of positive characteristic?
• RQ2 How can the Cartwright-Steger lattice $\Gamma$ be used to produce finite quotients that yield Ramanujan complexes via congruence subgroups?
• RQ3 What is the explicit set of generators for $\mathrm{PGL}_d(\mathbb{F}_{q^e})$ such that the associated Cayley complex is Ramanujan and covered by the Bruhat-Tits building?
• RQ4 Can the construction be refined to yield finite quotients isomorphic to subgroups of $\mathrm{PGL}_d(L)$ containing $\mathrm{PSL}_d(L)$ for finite local rings $L$?
• RQ5 What is the algorithmic procedure for computing the explicit $d \times d$ matrix generators of $\Gamma$ over $\mathbb{F}_q[x]$?

Key findings

• For every $d \geq 2$, $q$ a prime power, and $e \geq 1$ (with $e > 1$ if $q = 2$), the group $G = \mathrm{PGL}_d(\mathbb{F}_{q^e})$ admits an explicit set $S$ of $\sum_{k=1}^{d-1} \genfrac{[}{]}{0pt}{2}{d}{k}_q$ generators such that the Cayley complex of $G$ with respect to $S$ is a Ramanujan complex.
• The Cayley complex is covered by the Bruhat-Tits building $\mathcal{B}_d(F)$ for $F = \mathbb{F}_q((y))$, and its $1$-skeleton is the Cayley graph $\mathrm{Cay}(G; S)$, with edges defined by the generators.
• The generators are explicitly given as $d \times d$ matrices over $\mathbb{F}_q[x]$, and the construction is algorithmically realizable, with a full example provided in Section 10.
• The finite quotients $\Gamma(I) \backslash \mathcal{B}_d(F)$ are isomorphic to Cayley complexes of subgroups of $\mathrm{PGL}_d(L)$ containing $\mathrm{PSL}_d(L)$, where $L$ is a finite local ring.
• The Ramanujan property of the complexes is established via the spectral theory of $\mathrm{PGL}_d(F)$-actions, relying on the global Jacquet-Langlands correspondence for function fields, which is assumed to hold in the function field setting.
• The construction generalizes to $r$-partite complexes for each $r$ dividing $d$, by considering subgroups of $\mathrm{PGL}_d(\mathbb{F}_{q^e})$ containing $\mathrm{PSL}_d(\mathbb{F}_{q^e})$, extending the $r$-partite structure known for Ramanujan graphs in the $d=2$ case.