New answers tagged geometric-group-theory
2
votes
Fundamental domain for the action on curve complex
As Andy says, that the action is co-compact follows from the "change of coordinates principle". Finding a fundamental domain for $S_2$ (the compact, connected, oriented, boundary-less ...
- 26.4k
5
votes
Accepted
When the fundamental group of subgraph of groups embeds?
As mentioned in comments, if $\mathcal{H}$ is a subgraph of a graph of groups $\mathcal{G}$, with the natural induced structure, then the map
$H=\pi_1(\mathcal{H})\to G=\pi_1(\mathcal{G})$
induced by ...
- 24.1k
4
votes
Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
Even for coarse maps between Gromov-hyperbolic spaces $f: X\to Y$ there are neither reasonable upper nor lower bounds of the type
$$
\psi_-((x,y)_z)\le (f(x), f(y))_{f(z)}\le \psi_+((x,y)_z)
$$
(where ...
- 9,748
3
votes
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
5
votes
Green's kernel estimates on finitely generated groups
Symmetric random walks on finitely generated groups of growth at most quadratic are recurrent, therefore the Green kernel
$$ \Theta(x)=\sum_{n\ge 0}p^{(n)}(x)$$
is infinite for all $x$. (The right-...
- 770
6
votes
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
5
votes
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
4
votes
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 ...
- 26.4k
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