Groups with infinitely many ends are not fraction groups

TL;DR

The paper proves that finitely generated groups with infinitely many ends cannot be expressed as fractions of proper subsemigroups, resolving a conjecture and providing new insights into the structure of free groups.

Contribution

It establishes that groups with infinitely many ends are not fraction groups of proper subsemigroups, solving Navas' conjecture and offering a new proof regarding free groups and orderings.

Findings

Finitely generated groups with infinitely many ends are not fraction groups.

Resolved Navas' conjecture positively.

Provided a new proof that free groups lack isolated left-invariant orderings.

Abstract

We show that any finitely generated group $F$ with infinitely many ends is not a group of fractions of any finitely generated proper subsemigroup $P$, that is $F$ cannot be expressed as a product $P P^{-1}$. In particular this solves a conjecture of Navas in the positive. As a corollary we obtain a new proof of the fact that finitely generated free groups do not admit isolated left-invariant orderings.