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

Remark: It has to be non compact.

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#
group theory

# Not so hyperbolic

# No Eckmann-Hilton

# Rank isn’t like dimension

# Outside in

# Spooky algebra II

# On primitive roots of unity

# Freedom

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

Remark: It has to be non compact.

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

1) A discrete normal subgroup of a connected topological group is central.

2) If is the universal covering, the total space admits a topological group structure such that is a group morphism. The kernel of is then isomorphic to the fundamental group of .

Let be the free group in the generators and . Consider the group morphism defined by . Prove that is a free group of infinite rank. (Hint: Think topologically!)

Given a group , embed in a larger group such that any automorphism of is the restriction of an inner automorphism of .

Exhibit a group , a set of generators for and a subgroup such that for all and yet is not normal in .

Finite subgroups of the group of units of a field are cyclic.

Every morphism of abelian groups vanishing on is identically zero.

Use this fact to prove that is not a free abelian group.