### Abstract

Let G be a real reductive algebraic group with maximal compact subgroup K, and let F-r be a rank r free group. We show that the space of closed orbits in Hom(F-r, G)/G admits a strong deformation retraction to the orbit space Hom(F-r, K)/K. In particular, all such spaces have the same homotopy type. We compute the Poincare polynomials of these spaces for some low rank groups G, such as Sp(4, IR) and U(2, 2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

Original language | English |
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Pages (from-to) | 275-294 |

Number of pages | 20 |

Journal | Forum mathematicum |

Volume | 28 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2016 |

### Keywords

- Character varieties
- real reductive groups
- representation varieties
- CHARACTER VARIETIES
- 2X2 MATRICES
- 3-MANIFOLDS
- INVARIANTS
- SPLITTINGS
- SURFACE

## Cite this

Casimiro, A., Florentino, C., Lawton, S., & Oliveira, A. (2016). Topology of moduli spaces of free group representations in real reductive groups.

*Forum mathematicum*,*28*(2), 275-294. https://doi.org/10.1515/forum-2014-0049