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A three-manifold is a space that locally looks like Euclidean three-dimensional space

8 votes
Accepted

Hyperbolic manifolds with infinite cyclic fundamental group

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 …
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10 votes

Hyperbolic $3$-manifold groups that embed in compact Lie groups

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 …
YCor's user avatar
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7 votes

Non-orientable 3-manifolds

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 …
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4 votes
Accepted

What is "topology in dimension 3.5"?

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

Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?

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 …
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11 votes
Accepted

Quasi-isometric rigidity of Nil

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 …
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