## Tesi etd-10142009-110048 |

Thesis type

Tesi di laurea specialistica

Author

PICOCO, NICOLA

URN

etd-10142009-110048

Title

The space of characters: some properties and examples

Struttura

SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di studi

MATEMATICA

Supervisors

**relatore**Prof. Costantino, Francesco

Parole chiave

- space of representations
- space of characters

Data inizio appello

30/10/2009;

Consultabilità

Completa

Riassunto analitico

The purpose of this thesis is to give a presentation of the space of characters of a finitely generated group G. This is obtained by considering first of all the representations of the group in SL(2,C): this gives a space, called the space of representations, that is easily shown to be a closed algebraic set in C^{4n}, where n is the number of generators of the group G. For every element g in G, one can consider the function on R(G) which associates to every point (a representation) the trace of the image of g under the representation corresponding to the point.

We will show that the ring T formed by all these functions is finitely-generated: the image of the application from R(G) to C^k obtained by the functions that generate T (for a suitable k), form the "space of characters.

Also in this case the so-obtained space turns out to be a closed algebraic set. However the proof is decidedly more complicated in comparison to the case of the space of the representations. In this thesis we follow the proof given by Culler-Shalen, in which many results from algebraic geometry are used. These results are recalled in the first chapter where in particular we show that all the rational maps from a smooth curve to a projective variety are regular (this result is based on Zariski's Theorem); in the same chapter we prove that every planar curve admits a desingularization. These two results will be the key to prove in chapter 2 that the character space is actually a closed affine variety.

The second chapter is devoted to defining the space of representations and the space of characters as well as to prove that the second is a closed algebraic variety. Besides the already cited results coming from algebraic geometry, we will need the Burnside lemma and Wedderburn's theorem. Both results will be proved in subsections of the same chapter.

In the same chapter we will comment on the relations between the space of representations and the space of characters.

In the third chapter we will provide explicit examples of spaces of characters of groups and of surfaces (i.e. of fundamental groups of surfaces) by providing a set of defining equations.

We will first treat the case of the circle and of the bouquet of two circles. Then we will analyze the spaces of characters of some 2-dimensional surfaces like the projective plane, the punctured torus, the torus and the Klein bottle.

We will show that the ring T formed by all these functions is finitely-generated: the image of the application from R(G) to C^k obtained by the functions that generate T (for a suitable k), form the "space of characters.

Also in this case the so-obtained space turns out to be a closed algebraic set. However the proof is decidedly more complicated in comparison to the case of the space of the representations. In this thesis we follow the proof given by Culler-Shalen, in which many results from algebraic geometry are used. These results are recalled in the first chapter where in particular we show that all the rational maps from a smooth curve to a projective variety are regular (this result is based on Zariski's Theorem); in the same chapter we prove that every planar curve admits a desingularization. These two results will be the key to prove in chapter 2 that the character space is actually a closed affine variety.

The second chapter is devoted to defining the space of representations and the space of characters as well as to prove that the second is a closed algebraic variety. Besides the already cited results coming from algebraic geometry, we will need the Burnside lemma and Wedderburn's theorem. Both results will be proved in subsections of the same chapter.

In the same chapter we will comment on the relations between the space of representations and the space of characters.

In the third chapter we will provide explicit examples of spaces of characters of groups and of surfaces (i.e. of fundamental groups of surfaces) by providing a set of defining equations.

We will first treat the case of the circle and of the bouquet of two circles. Then we will analyze the spaces of characters of some 2-dimensional surfaces like the projective plane, the punctured torus, the torus and the Klein bottle.

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