Question: What Lie group acts transitively on the mapping torus of $ \begin{bmatrix} -1 & -1 \\ 0 & -1 \end{bmatrix} $ (which is exactly the negative of the mapping class of a Dehn twist)?
Some background:
Let $ N_1 $ be the Heisenberg nilmanifold (the total space of the unique principal circle bundle over the two torus $ T^2 $ with Euler number $1$). The Heisenberg nilmanifold is often constructed as the quotient of the group of three by three upper triangular matrices (also known as the Heisenberg group) $$ UT(3, \mathbb{R}) = \left\{\begin{bmatrix} 1 & x & z\\ 0 & 1 & y\\ 0 & 0 & 1\end{bmatrix} : x, y, z \in \mathbb{R}\right\} $$ by the subgroup of matrices with integer entries $$ UT(3, \mathbb{Z})=\Gamma_1 = \left\{\begin{bmatrix} 1 & n & k \\ 0 & 1 & m\\ 0 & 0 & 1\end{bmatrix} : n, m, k \in \mathbb{Z}\right\} $$ The Heisenberg Nilmanifold coincides with the mapping torus of the Dehn twist of the two torus $ T^2 $ $$ N_1\cong MT\Bigg(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\Bigg) \cong UT(3, \mathbb{R})/UT(3, \mathbb{Z}) $$ The Heisenberg Nilmanifold is double covered by the manifold $ N_2 $, the principal circle bundle over $ T^2 $ with Euler number 2. $ N_2 $ coincides with the mapping torus of two Dehn twists. $ N_2 $ can also be realized as the quotient $ UT(3, \mathbb{R}) $ by $$ \Gamma_2 = \left\{\begin{bmatrix} 1 & n & \frac{k}{2} \\ 0 & 1 & m\\ 0 & 0 & 1\end{bmatrix} : n, m, k \in \mathbb{Z}\right\} $$ In other words $$ N_2\cong MT\Bigg(\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\Bigg) \cong UT(3, \mathbb{R})/\Gamma_2 $$ There is a third manifold, also a mapping torus of $ T^2 $, which I will call $ \mathcal{N}_1 $ $$ \mathcal{N}_1:= MT\Bigg( \begin{bmatrix} -1 & -1 \\ 0 & -1 \end{bmatrix} \Bigg) $$ Like the Heisenberg nilmanifold, $ \mathcal{N}_1 $ is a three manifold admitting Nil geometry and double covered by $ N_2 $. And this question
https://mathoverflow.net/questions/416611/torus-bundles-and-compact-solvmanifolds
shows that every mapping torus of $ T^2 $ is a solvmanifold. Thus some solvable group $ G $ acts transitively on $ \mathcal{N}_1 $. Despite the fact that $ \mathcal{N}_1 $ and the Heisenberg nilmanifold both admit transitive actions by solvable groups and both are three manifolds admitting Nil geometry and are double covered by $ N_2 $ it turns out that $ \mathcal{N}_1 $ is topologically distinct from the Heisenberg nilmanifold (see Is every Nil manifold a nilmanifold? for a justification of this fact using Seifert fibrations or for a more substantial proof arguing why in this case mapping classes which are not conjugates or inverses in the mapping class group must give distinct total spaces see here Are these mapping tori different?).
Moreover it seems that even though $ UT(3, \mathbb{R}) $ acts transitively on both $ N_1 $ and $ N_2 $ it cannot on $ \mathcal{N}_1 $.
So my question is: What group acts transitively on $ \mathcal{N}_1=MT\Bigg( \begin{bmatrix} -1 & -1 \\ 0 & -1 \end{bmatrix} \Bigg) $ ?
