# A question about the first $$L^2$$-Betti number

In a recent arXiv preprint (arxiv:2106.15750) J. A. Hillman discusses a new homological approach towards some old results on 3-manifold groups due to Elkalla. In his article he runs into an interesting question concerning the first $$L^2$$-Betti number of finitely generated groups:

Assume that a finitely generated group G has infinite subgroups $$N\leq U$$ such that $$N$$ is subnormal in $$G$$, $$U$$ is finitely generated and the index $$[G:U]$$ is infinite. Does the first $$L^2$$-Betti number $$\beta_1^{(2)}(G)$$ vanish.

It is a result of Gaboriau (https://eudml.org/doc/104184) that if $$G$$ has an infinite normal subgroup $$N$$ of infinite index, which is finitely generated, then $$\beta_1^{(2)}(G) = 0$$. So, the above question asks for a vast (?) generalization of that result.