A fundamental domain for $PGL(2,\mathbb{F}_q[t])\backslash PGL\!\left(2,\mathbb{F}_q\!\left(\!(t^{-1})\!\right)\!\right)$

TL;DR

This paper constructs a fundamental domain for the action of PGL(2, F_q[t]) on PGL(2, F_q((t^{-1}))) by explicitly describing a subset of ordered triples in the projective line over the local field.

Contribution

It provides a strong, explicit fundamental domain for the quotient space of PGL(2) over a local function field by a polynomial subgroup, advancing understanding of their geometric structure.

Findings

Explicit description of the fundamental domain as a subset of ordered triples.

Clarification of the quotient's geometric structure in the context of local function fields.

Enhanced understanding of the action of PGL(2, F_q[t]) on the associated projective line.

Abstract

We give a strong fundamental domain for the quotient of $PGL_2\!\left(\mathbb{F}_q(\!(t^{-1})\!)\!\right)$ by $PGL_2\!\left(\mathbb{F}_q[t]\right)$ as a subset of distinct ordered triple points of $\mathbb{P}^1(\mathbb{F}_q(\!(t^{-1})\!))$.