Does the group
$$
G
=
\left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\}
$$
have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$.
After googling a few variantions of "matrices with unimodular determinant" and coming up empty, I thought that I would see what people here think.
Edit: Notice that $G$ is strictly larger than both $U(n)$ and $SL(n)$. Consider $$ A=\begin{pmatrix} 1 & 1 \\ 0 & -1 \end{pmatrix}. $$ We have $A \in G$, but $A$ is not unitary and $A \notin SL(2,\mathbb C)$.
Even though $G \neq SL(n,\mathbb C)$, they are related in the sense that $S^1 \times SL(n,\mathbb C) $ is an $n$-fold cover of $G$ -- the covering is defined by $(z,A) \mapsto z \cdot A$. (This is an $n$-to-1 map because $\det(zA) = z^n \det(A)$ for $n \times n$ matrices $A$).