The Geometry of Rank 2 Hyperbolic Root Systems

Let ∆ be a rank 2 hyperbolic root system. Then ∆ has generalized Cartan matrix H(a, b) = ( 2 −b −a 2 ) indexed by a, b ∈ Z with ab ≥ 5. If a 6= b, then ∆ is non-symmetric and is generated by one long simple root and one short simple root, whereas if a = b, ∆ is symmetric and is generated by two long simple roots. We prove that if a 6= b, then ∆ contains an infinite family of symmetric rank 2 hyperbolic root subsystems H(k, k) for certain k ≥ 3, generated by either two short or two long simple roots. We also prove that ∆ contains non-symmetric rank 2 hyperbolic root subsystems H(a′, b′), for certain a′, b′ ∈ Z with a′b′ ≥ 5.