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.

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