Ternary forms as invariants of Markoff forms and other SL 2 (Z)-bundles

Abstract

Consider the free group Γ = { A, B} generated by matrices A, B in SL 2( Z). We can construct a ternary form Φ( x, y, z) whose GL 3( Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL 2( Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = - z 2+ y(2 x+ y+ z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.