Ihara’s theorem

Let Γ be a discrete, torsion-free subgroupMathworldPlanetmathPlanetmath of SL2p (where p is the field of p-adic numbers (http://planetmath.org/PAdicIntegers)). Then Γ is free.

Proof, or a sketch thereof.

There exists a p+1 regularPlanetmathPlanetmath tree X on which SL2p acts, with stabilizerMathworldPlanetmath SL2p (here, p denotes the ring of p-adic integers (http://planetmath.org/PAdicIntegers)). Since p is compact in its profinite topology, so is SL2p. Thus, SL2pΓ must be compact, discrete and torsion-free. Since compact and discrete implies finite, the only such group is trivial. Thus, Γ acts freely on X. Since groups acting freely on trees are free, Γ is free. ∎

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