Computation of some leafwise cohomology ring

TL;DR

This paper computes the leafwise cohomology ring of a specific foliation on a quotient space of SL(2,R) using non-abelian harmonic analysis, providing new insights into the topology of these foliations.

Contribution

It introduces a method to explicitly compute the leafwise cohomology ring for orbit foliations associated with SL(2,R) and parabolic subgroups, advancing understanding of their geometric structure.

Findings

Explicit computation of the leafwise cohomology ring $H^*(F_P)$

Application of non-abelian harmonic analysis techniques

Enhanced understanding of the topology of orbit foliations

Abstract

Let $G$ be the group $SL(2,\mathbb{R})$, $P\subset G$ be the parabolic subgroup of upper triangular matrices and $\Gamma\subset G$ be a cocompact lattice. A right action of $P$ on $\Gamma\backslash G$ defines an orbit foliation $\mathcal{F}_P$. We compute the leafwise cohomology ring $H^*(\mathcal{F}_P)$ by exploiting non-abelian harmonic analysis on $G$.