# Tagged Questions

**4**

votes

**1**answer

227 views

### What is known about maximal free subgroups of surface groups?

Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...

**6**

votes

**3**answers

399 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 ...

**0**

votes

**0**answers

60 views

### Universal covering of a submanifold with boundary

Good afternoon,
I'm having a hard time trying to prove the following lemma :
Let $V$ be a hyperbolic compact orientable manifold, containing two totally geodesic com pact hypersurfaces $H$ and $G$ ...

**11**

votes

**2**answers

457 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 ...

**8**

votes

**0**answers

215 views

### Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$

Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...

**4**

votes

**2**answers

249 views

### Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for ...

**2**

votes

**1**answer

645 views

### How to rigorously prove that simple closed curves on a surface are primitive closed curves ?

Let me first state the definitions :
A not-nullhomotopic closed curve / loop $c$ on an orientable surface $X,c:[0,1]\to X$ is called simple closed curve is $c|[0,1)$ is injective and [ $c(0)=c(1) ] ...

**3**

votes

**2**answers

462 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.
...

**0**

votes

**1**answer

190 views

### altering curvature on a tessellation representation of a compact surface

I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...