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Ramanujan Complexes from Unitary Groups over Number Fields

Rahul Dalal, Alberto Mínguez, Jiandi Zou

Abstract

In this article, we construct new families of Ramanujan complexes with local structure distinct from all previously known examples. Our approach is based on unitary groups over number fields, more specifically on what we call super-definite unitary groups, that is definite unitary groups that are anisotropic modulo their center at a finite place. These arise naturally as groups of units in central division algebras with involution of the second kind. Our first main result gives a general construction of infinite families of Ramanujan complexes associated with a super-definite unitary group $G$ over a totally real number field and a finite place $v_0$. The structure of the resulting complex is governed by the type of the Bruhat-Tits building at $v_0$. It includes new examples of type $A_n$ when $v_0$ is split, and novel families of type ${}^2\!A'_n$, ${}^2 \! A''_n$ (with $n$ even), $B$-$C_n$, ${}^2 \! B$-$C_n$ and $C$-$BC_n$ in the non-split case. This construction works uniformly across all ranks. Since much of the motivation for constructing expander complexes comes from computer science, we investigate the algorithmic explicitness of our construction in the latter part of the paper, and provide an example in rank 5 where it becomes fully explicit. In particular, this example yields golden gates for the real Lie group $PU(5)$.

Ramanujan Complexes from Unitary Groups over Number Fields

Abstract

In this article, we construct new families of Ramanujan complexes with local structure distinct from all previously known examples. Our approach is based on unitary groups over number fields, more specifically on what we call super-definite unitary groups, that is definite unitary groups that are anisotropic modulo their center at a finite place. These arise naturally as groups of units in central division algebras with involution of the second kind. Our first main result gives a general construction of infinite families of Ramanujan complexes associated with a super-definite unitary group over a totally real number field and a finite place . The structure of the resulting complex is governed by the type of the Bruhat-Tits building at . It includes new examples of type when is split, and novel families of type , (with even), -, - and - in the non-split case. This construction works uniformly across all ranks. Since much of the motivation for constructing expander complexes comes from computer science, we investigate the algorithmic explicitness of our construction in the latter part of the paper, and provide an example in rank 5 where it becomes fully explicit. In particular, this example yields golden gates for the real Lie group .
Paper Structure (75 sections, 26 theorems, 191 equations, 2 figures)

This paper contains 75 sections, 26 theorems, 191 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context and Motivation
  3. Our Contribution
  4. Explicitness
  5. Possible Extensions
  6. Conditionality
  7. Acknowledgments
  8. Notation
  9. Conventions
  10. Notation Reference
  11. General Theory
  12. Background Material
  13. Complexes and Buildings
  14. Simplicial Complexes
  15. Quotients of Complexes
  16. ...and 60 more sections

Key Result

Theorem 1.2.1

Let $p \neq 2,7$ be prime and let $\mathcal{B}_p$ be the Bruhat-Tits building for Then we give an explicit algorithm which, given any $n$ relatively prime to $2 \cdot 7 \cdot p$ produces a Ramanujan Complex $\mathcal{X}_n$ with universal cover $\mathcal{B}_p$. The algorithm runs in time polynomial in $n$ and the size of $\mathcal{X}_n$ grows polynomially in $n$.

Figures (2)

  • Figure 1: Affine Dynkin Diagram of type $\mathrm{C}$-$\mathrm{BC}_2^{\mathrm{IV}}$ and $\mathrm{C}$-$\mathrm{BC}_2^{\mathrm{III}}$
  • Figure 2: Apartment and chamber of type ${}^2 \! A'_4$

Theorems & Definitions (67)

  • Theorem 1.2.1
  • Claim 1.2.2: Informal
  • Definition 2.1.1
  • Example
  • Proposition 2.1.2
  • Proposition 2.1.3
  • Definition 2.1.4
  • Remark 2.1.6
  • Remark 2.1.7
  • Definition 2.2.1: Lub18
  • ...and 57 more