# Tag Info

Accepted

• 26.3k

• 26.3k
Accepted

### Identifying two definitions of orientation on a vector space

Here's a direct way to relate the two: One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing  \Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n+...
• 8,624

### Residual finiteness of hyperbolic 3-manifold groups

The answer to Q1 is negative in general (allowing infinitely generated fundamental group). See Example 2 which is a discrete torsion-free subgroup $G< PSL_2(\mathbb{C})$, hence $\mathbb{H}^3/G$ is ...
• 66.8k
Accepted

### Simple convergence of convex compact set implies Hausdorff convergence

A counterexample is given by $n=2$, $C=\{(0,0)\}$, and $C_k=\{(t,kt)\colon0\le t\le1\}$ (or one can instead take $C_k=\{(t,t/k)\colon0\le t\le1\}$). Another counterexample, in the same spirit, is ...
• 117k

### Residual finiteness of hyperbolic 3-manifold groups

Here's another negative answer for Q2. I'm assuming (as in Sam Nead's answer) that the covering should be locally isometric. By Ahlfors-Bers, a tame infinite volume hyperbolic manifold with ends of ...
• 19.2k

### Residual finiteness of hyperbolic 3-manifold groups

Sam Nead's answer does it, but perhaps I can offer a slightly different perspective on Question 1. No complicated hyperbolic gluing results are needed. I assume we are satisfied with the ...
• 24.1k

### Residual finiteness of hyperbolic 3-manifold groups

The answer to the first question is "yes" and the answer to the second is "no", assuming that you are looking for a covering which is a locally isometric. If you do not require a ...
• 26.3k
Accepted

### Hyperbolic three-manifolds that fiber over the circle

Question: Is the length of homotopically non-trivial loops in $M(f)$ bounded below in terms of the genus $g$ of the fiber? Answer: No. Here is one family of examples. Suppose that $S$ is the closed ...
• 26.3k
1 vote

### Space of the trivial long knot in the thickened surface

Let us show that $\mathcal E=Emb_0(I,F\times I)\sim\Omega_0(F,x_0)$. We start with R. Budney's remark. Proposition. Let $F$ be a connected compact 2-manifold and $P(F)$ the pseudoisotopy group, i.e ...
• 357

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