## Thesis etd-10092017-232207 |

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Thesis type

Tesi di laurea magistrale

Author

BARBERIS, MARCO

URN

etd-10092017-232207

Thesis title

Q.i. rigidity for lattices of hyperbolic isometries.

Department

MATEMATICA

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Frigerio, Roberto

Keywords

- Geometric group theory
- Hyperbolic geometry
- Hyperbolic lattices
- Mostow Rigidity Theorem
- Quasiconformal maps
- Thick-thin decomposition
- Uniformly quasiconformal groups

Graduation session start date

27/10/2017

Availability

Full

Summary

Given a finitely generated group, a natural metric on it, arising just from its algebraic structure, is well defined up to quasi isometry. It is easy to check that two virtually isomorphic groups are indeed quasi isometric, but the converse is not true. Due to this fact we say that a class of group is quasi isometrically rigid if any group quasi isometric to a group in the class is virtually isomorphic to a possibly different member of the class.

In this thesis we first prove that in dimension at least 3 the class of uniform hyperbolic lattices, i.e. discrete subgroups of isometries of the hyperbolic space with compact quotient, is quasi isometrically rigid, as proved by Pekka Tukia in 1986. Moreover we prove that every nonuniform hyperbolic lattice, i.e. a non cocompact discrete subgroup of hyperbolic isometries with finite-volume quotient space, is quasi isometrically rigid by itself, following the original proof given by Richard Schwartz in 1996. We conclude the thesis by using the previously explained methods in order to prove a classical theorem in hyperbolic geometry: Mostow Rigidity Theorem.

In this thesis we first prove that in dimension at least 3 the class of uniform hyperbolic lattices, i.e. discrete subgroups of isometries of the hyperbolic space with compact quotient, is quasi isometrically rigid, as proved by Pekka Tukia in 1986. Moreover we prove that every nonuniform hyperbolic lattice, i.e. a non cocompact discrete subgroup of hyperbolic isometries with finite-volume quotient space, is quasi isometrically rigid by itself, following the original proof given by Richard Schwartz in 1996. We conclude the thesis by using the previously explained methods in order to prove a classical theorem in hyperbolic geometry: Mostow Rigidity Theorem.

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