This movie shows the region of the 3 x-coordinates as the 2 y-coordinates move along a diagonal in the Grenier fundamental domain for GL(3,Z) in the determinant 1 surface. The domain is described on page 151 of the old edition of Harmonic Analysis on Symmetric Spaces and Applications, Volume II. The Mathematica program to create the movie is as follows.

haro[t_,w_]:=haro[t,w]=RegionPlot3D[(1-x1+x2)^2+t^2*((1-x3)^2+w^2)>=1

&&(x1-x2)^2+t^2*((1-x3)^2+w^2)>=1

&&x1^2+t^2>=1

&&x2^2+t^2*(x3^2+w^2)>=1

&&x3^2+w^2>=1,

{x1,0,.5},{x2,-.5,.5},{x3,0,.5},

Mesh->None,PlotStyle->Directive[Opacity[0.5],Pink,Specularity[White,20]],PlotPoints->60]

Export["grenier.gif",Table[haro[w,w],{w,.9,1.03,.005}]]

grenier.gif

Import["grenier.gif", "Animation"]