The binary space partition (BSP) is a recursive cutting scheme for aset of n disjoint polygonal scenes in the Euclidean space. We split thespace along a plane into two parts and then we partition recursivelyboth parts until the interior of every space fragment is disjointfrom the polygonal scenes. The size of a BSP is the number of splitsmade. The efficiency of applications (in computer graphics andcomputational geometry) vitally depends on the size of the BSPs.Paterson and Yao proved that the size of the smallest BSP for nrectangles is $Theta(n^2)$ in the worst case; and $Theta(n^{3/2})$ if allrectangles are orthogonal. Later, Agarwal et al. showed that for northogonal fat rectangles there is a BSP of size n $2^{(log n)^{1/2}}$.(A rectangle is fat if its aspect ratio is bounded by a constant.) Inthis talk, I show that one can generate a BSP of size $O(n log^8 n)$ forn orthogonal fat rectangles, improving the bound of Agarwal et al.,using the new technique of overlays of BSP

We explore the group of 2x2 determinant 1 matrices with coefficients inthe ring of integers Z as well as an analog obtained by replacing Z by thering of polynomials F[t] where F is a finite field. Decomposition theoremsfor these groups will be obtained using the actions of the groups ontrees, which are connected graphs having no cycles, such as that of degree3 a part of which is pictured above.REFRESHMENTS WILL BE SERVED!