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NON-ARITHMETIC UNIFORMIZATION OF SOME REAL MODULI SPACES
 

Summary: NON-ARITHMETIC 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 non-arithmetic group. The whole moduli
space is made from some incommensurable arithmetic pieces, in
the spirit of the construction of Gromov and Piatetski-Shapiro.
1. Introduction
The purpose of this paper is to explain how some real moduli spaces
have non-arithmetic uniformizations, in the sense that they are homeo-
morphic to real hyperbolic space modulo the action of a non-arithmetic
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 non-arithmetic groups by Gromov and Piatetski-Shapiro [6].
In other words, some real moduli spaces give very natural and con-
crete examples of the Gromov-Piatetski-Shapiro 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

  

Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics