Weak Ramanujan property of the standard non-uniform arithmetic quotient of PGL₄

2022/06/01 by Hong, Soonki, Kwon, Sanghoon · 1 citation
#20G25 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 20E42 #Secondary 47A25 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2206.00567

Abstract

Let F be a field of formal series over a finite field and B_d be the affine building associated to PGL_d(F). Given a lattice Γ in PGL_d(F), the complex arising as a quotient Γ\backslash B_d is called weakly Ramanujan if every non-tivial discrete simultaneous spectrum of the colored adjacency operators A₁,A₂,…,A_d-1 acting on L²(Γ\backslash B_d) is contained in the simultaneous spectrum of those operators acting on L²(B). In this paper, we prove that the standard non-uniform arithmetic quotient PGL₄(\mathbbF_q[t])\backslash B₄ of PGL₄(F) is weakly Ramanujan.

Cited by

• Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of PGL_d

Related

• Explicit Constructions of Ramanujan Complexes of Type A_d
• Ramanujan Complexes of Type A_d
• Highlights from "The Ramanujan Property for Simplicial Complexes" [arXiv:1605.02664]
• The Ramanujan Property for Simplicial Complexes
• Quasi-isometric embeddings of Ramanujan complexes