Sarnak’s conjecture on trace-set properties implying arithmeticity

Prove Sarnak’s conjecture for any cofinite Fuchsian group Γ: (i) Establish that if the trace set Tr(Γ) satisfies the bounded clustering property—meaning there exists a constant K>0 such that for every integer n, the cardinality of Tr(Γ)∩[n,n+1] is at most K—then Γ is arithmetic; and (ii) Establish that if the trace-set gap Gap(Tr(Γ)):=inf{|a−b|: a,b∈Tr(Γ), a≠b} is positive, then Γ is derived from a quaternion algebra.

Background

The paper studies trace sets of Fuchsian lattices and recalls well-known conjectures relating trace-set structure to arithmeticity. Luo and Sarnak previously showed that the trace set of an arithmetic Fuchsian group satisfies the bounded clustering property, and Sarnak conjectured the converse, adding a second statement about the presence of a positive gap in the trace set implying derivation from a quaternion algebra.

The authors note that while part of Sarnak’s conjecture has been confirmed for nonuniform Fuchsian lattices, the conjecture remains completely open in the cocompact case. Establishing the full conjecture would provide a definitive characterization linking combinatorial properties of traces to arithmeticity across all cofinite Fuchsian groups.