Non-commensurable hyperbolic manifolds with the same trace ring

TL;DR

This paper demonstrates the existence of infinitely many non-commensurable hyperbolic n-manifolds sharing the same trace ring and ambient group, with compact examples possible in dimensions four and higher.

Contribution

It establishes the surprising result that non-commensurable hyperbolic manifolds can have identical trace rings and ambient groups across all dimensions n ≥ 3.

Findings

Infinitely many non-commensurable hyperbolic manifolds with same trace ring.

Existence of compact examples for dimensions n ≥ 4.

Trace ring does not determine commensurability class.

Abstract

We prove that there are infinitely many pairwise non-commensurable hyperbolic $n$-manifolds that have the same ambient group and trace ring, for any $n \geq 3$. The manifolds can be chosen compact if $n \geq 4$.