ℓ2$\ell ^2$‐Betti numbers and coherence of random groups

Abstract

We study ℓ 2 $\ell ^2$ -Betti numbers, coherence and (virtual) fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have ℓ 2 $\ell ^2$ -homology concentrated in dimension 1 and embed in a virtually free-by-cyclic group with high probability. In the case of Euler characteristic zero, we use Novikov homology to show that a random group is free-by-cyclic with positive probability.