Realizing algebraic invariants of hyperbolic surfaces

TL;DR

This paper demonstrates that for any real number field and compatible quaternion algebra, there exists a hyperbolic surface of genus at least 2 whose invariant trace field and quaternion algebra match these algebraic structures, linking algebraic invariants to geometric structures.

Contribution

It constructs hyperbolic surfaces with prescribed algebraic invariants, specifically invariant trace fields and quaternion algebras, expanding the understanding of algebraic invariants in hyperbolic geometry.

Findings

Existence of hyperbolic structures with given invariant trace fields.

Existence of hyperbolic structures with prescribed quaternion algebras.

Link between algebraic invariants and geometric realizations.

Abstract

Let $S_g$ ($g\geq 2$) be a closed surface of genus $g$. Let $K$ be any real number field and $A$ be any quaternion algebra over $K$ such that $A\otimes_K\mathbb{R}\cong M_2(\mathbb{R})$. We show that there exists a hyperbolic structure on $S_g$ such that $K$ and $A$ arise as its invariant trace field and invariant quaternion algebra.