# Quadratic Thurston maps with few postcritical points

## TL;DR

This paper classifies all non-exceptional quadratic Thurston maps with fewer than five postcritical points using self-similar group theory, linking combinatorial classes to quadratic rational maps with fewer than four postcritical points.

## Contribution

It provides a complete enumeration of such Thurston maps and connects their combinatorial classes to specific quadratic rational maps.

## Key findings

- All combinatorial classes of non-exceptional quadratic Thurston maps with <5 postcritical points are enumerated.
- Maps on moduli space can be realized by quadratic rational maps with <4 postcritical points.
- The enumeration leverages the theory of self-similar groups.

## Abstract

We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on moduli space can be realized by quadratic rational maps with fewer than four postcritical points.