All Questions
Tagged with fundamental-group hyperbolic-geometry
9 questions
11
votes
1
answer
270
views
Example of three dimensional atoroidal Poincaré duality group with some pathology
I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...
- 1,638
6
votes
1
answer
336
views
Geodesic representatives in the orbifold fundamental group
Does every element in the orbifold fundamental group $\pi_1^{orb}(X,x)$ of a closed hyperbolic 2-orbifold $X$ admit a unique geodesic arc representing it?
Does every free homotopy class in $X$ admit ...
- 73
12
votes
1
answer
798
views
Finite covers of hyperbolic surfaces and the `second systole´
We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: injectivity radius of hyperbolic ...
- 3,928
6
votes
1
answer
405
views
Fundamental groups of hyperbolic $4$-manifolds and $\rm CAT(0)$ cube complexes
Suppose $M^4$ is a compact hyperbolic (i.e. curvature $-1$) $4$-manifold and $\Gamma\cong\pi_1(M^4)$.
Is there any expectation whether $\Gamma$ acts properly and co-compactly on a $\rm CAT(0)$ cube ...
- 14.3k
8
votes
1
answer
573
views
Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
- 103
8
votes
3
answers
943
views
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
- 81
16
votes
2
answers
3k
views
The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
- 20.9k
9
votes
1
answer
1k
views
Fundamental group of an hyperbolic $4$-manifold
Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...
- 2,696
4
votes
2
answers
1k
views
Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable
I believe the following statement is true, and I've even seen it referenced here. Could someone point me to a proof?
The fundamental group of a closed hyperbolic 3-manifold is not a free product.
- 726