Is SL(2,C) unimodular

Question

My question is in the title, do you know if SL(2,C) is an unimodular group or not and how to prove it ?

Thank you.

Answer 1

I found the answer in the article of JÖRG WINKELMANN : "Complex analytic geometry of complex parallelizable manifolds" :

Recall that a locally compact topological group $G$ is called unimodular iff a left-invariant Haar measure is also right-invariant.

Assume $G$ is a locally compact topological group.

Page $5$, J. Winkelmann proved that :

• Let $\Gamma$ be a cocompact discrete subgroup of $G$. Then $G/\Gamma$ admits a $G$ invariant probability measure.

• Let $\Gamma$ be a discrete subgroup of $G$. Assume that there exists a $G$-invariant probability measure on $G/\Gamma$ then $G$ is unimodular.

Finally, the existence of lattices in $SL_2(\mathbb{C})$ is given by the existence of compact $3$-dimensional hyperbolic manifold and thus we know that $SL_2(\mathbb{C})$ is unimodular.

I hope I did not make mistakes.