# Ihara’s theorem

Let $\mathrm{\Gamma}$ be a discrete, torsion-free subgroup^{} of ${\mathrm{SL}}_{2}{\mathbb{Q}}_{p}$ (where ${\mathbb{Q}}_{p}$ is the field of $p$-adic numbers (http://planetmath.org/PAdicIntegers)). Then $\mathrm{\Gamma}$ is free.

###### Proof, or a sketch thereof.

There exists a $p+1$ regular^{} tree $X$ on which ${\mathrm{SL}}_{2}{\mathbb{Q}}_{p}$ acts, with stabilizer^{} ${\mathrm{SL}}_{2}{\mathbb{Z}}_{p}$ (here, ${\mathbb{Z}}_{p}$ denotes the ring of $p$-adic integers (http://planetmath.org/PAdicIntegers)). Since ${\mathbb{Z}}_{p}$ is compact in its profinite topology, so is ${\mathrm{SL}}_{2}{\mathbb{Z}}_{p}$. Thus, ${\mathrm{SL}}_{2}{\mathbb{Z}}_{p}\cap \mathrm{\Gamma}$ must be compact, discrete and torsion-free. Since compact and discrete implies finite, the only such group is trivial. Thus, $\mathrm{\Gamma}$ acts freely on $X$. Since groups acting freely on trees are free, $\mathrm{\Gamma}$ is free.
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Title | Ihara’s theorem |
---|---|

Canonical name | IharasTheorem |

Date of creation | 2013-03-22 13:54:26 |

Last modified on | 2013-03-22 13:54:26 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 9 |

Author | bwebste (988) |

Entry type | Theorem |

Classification | msc 20G25 |