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Groups, Geometry, and Dynamics

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Volume 9, Issue 1, 2015, pp. 317–323
DOI: 10.4171/GGD/314

Published online: 2015-04-28

Groups with infinitely many ends are not fraction groups

Dawid Kielak[1]

(1) Universit├Ąt Bonn, Germany

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.

Keywords: Groups with infinitely many ends, groups of fractions, isolated orderings

Kielak Dawid: Groups with infinitely many ends are not fraction groups. Groups Geom. Dyn. 9 (2015), 317-323. doi: 10.4171/GGD/314