# Not so hyperbolic

Give an example of a hyperbolic manifold with finite volume and non hyperbolic fundamental group.

Remark: It has to be non compact.

# Bend over 2

Prove that a compact oriented surface of genus greater or equal to two has a hyperbolic structure, i.e. a metric with gaussian curvature constantly $-1$.

# Cover this

Let $f:M\to N$ be a local isometry between connected riemannian manifolds. If $M$ is complete, then $N$ is complete and $f$ is a covering map.

# No Eckmann-Hilton

Prove the following two statements and conclude that the fundamental group of a (connected) topological group $G$ is abelian:

1) A discrete normal subgroup of a connected topological group is central.
2) If $p:\hat{G}\to G$ is the universal covering, the total space $\hat{G}$ admits a topological group structure such that $p$ is a group morphism. The kernel of $p$ is then isomorphic to the fundamental group of $G$.

# Rank isn’t like dimension

Let $F_2(a,b)$ be the free group in the generators $a$ and $b$. Consider the group morphism $f: F_2(a,b) \to \mathbb{Z}$ defined by $f(a)=f(b)=1$. Prove that $Ker(f)$ is a free group of infinite rank. (Hint: Think topologically!)

# Spooky geometry V

The complement of an algebraic set in $A^n(\Bbb{C})$ is path-connected.

# Contractible manifolds

Classify the closed contractible manifolds (Recall that a manifold is closed if it is compact and its boundary is empty.)