 
Summary: NONARITHMETIC UNIFORMIZATION OF SOME
REAL MODULI SPACES
DANIEL ALLCOCK, JAMES A. CARLSON, AND DOMINGO TOLEDO
Abstract. Some real moduli spaces can be presented as real hy
perbolic space modulo a nonarithmetic group. The whole moduli
space is made from some incommensurable arithmetic pieces, in
the spirit of the construction of Gromov and PiatetskiShapiro.
1. Introduction
The purpose of this paper is to explain how some real moduli spaces
have nonarithmetic uniformizations, in the sense that they are homeo
morphic to real hyperbolic space modulo the action of a nonarithmetic
group. The space is assembled, in a natural way, from various pieces,
each of which can be uniformized by an arithmetic group. One can
check that the pieces are not all commensurable. The uniformization
of the moduli space can be seen as an orbifold version of the construc
tion of nonarithmetic groups by Gromov and PiatetskiShapiro [6].
In other words, some real moduli spaces give very natural and con
crete examples of the GromovPiatetskiShapiro construction. We first
found this phenomenom in the moduli space of real cubic surfaces [2],
[3]. Since some of the details there are quite technical, in this paper
