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For question D, I think the answer is negative. Consider the semidirect product $\Lambda=\mathbf{Z}^2\rtimes_A\mathbf{Z}$, with $A=\begin{pmatrix}25 & 7\\ 7 & 2\end{pmatrix}$. This is the $\pi_1$ of s …

This does not answer the question but at least gives a counterexample to the conjecture in Luo's paper. I assume that $\mathrm{PSL}_2(R)$ is defined as the quotient of $\mathrm{SL}_2(R)$ by its cente …

cw answer: As mentioned in the comments, this just refers to the relations between 3-dimensional and 4-dimensional topology.

Yes, there exists such closed hyperbolic (= constant curvature $-1$) manifolds with this property, in arbitrary dimension. For $d\ge 1$, let $q_t$ be a quadratic form of rank $d$ with coefficients in …

Here's a sketch of proof (which intersects yours): Let $\Gamma$ be QI to NIL. As you say, by Gromov's theorem, $\Gamma$ is virtually nilpotent; let some finite index subgroup be a lattice in some sim …

This consists in classifying non-elliptic elements of the Lie group $\mathrm{Isom}(\mathbf{H}^n)\simeq\mathrm{PO}(n,1)$ up to conjugacy and inversion. One can do separately loxodromics and horocyclic …