Unitary unit conjecture for group rings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:08:23Z http://mathoverflow.net/feeds/question/79702 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79702/unitary-unit-conjecture-for-group-rings Unitary unit conjecture for group rings Joerg Sixt 2011-11-01T11:47:22Z 2011-11-01T16:19:14Z <p>The famous "unit conjecture" for group rings states that all units of a group ring $K[G]$ are trivial for a field $K$ and a torison-free group $G$. We are far away from solving the conjecture (See e.g. <a href="http://people.maths.ox.ac.uk/craven/docs/seminars/220909slides.pdf" rel="nofollow">http://people.maths.ox.ac.uk/craven/docs/seminars/220909slides.pdf</a>). </p> <p>I wonder whether more is known about unitary units: Given a group homomorphism $w\colon G \rightarrow \mathbb{Z}/2\mathbb{Z}=\left\{\pm 1\right\}$ we can define an involution on $K[G]$ via $g\in G \mapsto \overline{g}=w(g)g^{-1}$ . The unitary units are those elements $x$ of the group ring with $\overline{x}x=1$. Do we know they are trivial for torsion-free groups?</p>