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

Abstract

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.