A space parametrizing isomorphism classes of geometric objects. Its topology and possible stabilizers are part of the definition; a moduli space of Riemann surfaces forgets the marking retained by Teichmüller space.
The moduli space of unmarked compact genus- Riemann surfaces. For it is the quotient of Teichmüller space by the mapping class group, with its quotient topology. The thick subsets are compact by Mumford's compactness theorem.
The subset of the unmarked moduli space of Riemann surfaces with hyperbolic systole at least is compact for fixed genus . Use a Bers pants decomposition theorem bound, finitely many topological decomposition types and Dehn twists to place representatives in finitely many compact boxes in Fenchel–Nielsen coordinates.
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In mathematics, particularly in algebraic geometry and differential geometry, a **moduli space** is a geometric space that parametrizes a family of algebraic structures, such as curves, vector bundles, or more generally, geometrical objects. The idea is to organize the objects of a particular type into a space, where each point in this space corresponds to a distinct structure (often up to some kind of equivalence).