HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

Abstract In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs ( P , τ), where is a compact Riemann surface with finite number holes and punctures τ: → an anti-holomorphic involution. We describe uniformisation curves by Fuchsian groups. construct their connected components. prove that any component homeomorphic to quotient finite-dimensional vector space discrete group determine dimensions these spaces.