I saw a paper that said it is well-known that for finitely generated group $G$: $\beta^{(2)}\_1(G) \le d(G)-1$, but I can't find any reference proving it.


$d(G)$ denotes the minimal number of generators of $G$.


Can you offer some references?



Reference proving $\beta^{(2)}_1(G) \le d(G)-1$
Here is an answer expanding on the comments of Uri Bader and Andy Putman. I think it is useful to prove this directly using the basic definitions and properties of the $L^2$-Betti numbers.


First, exclude the case that $G$ is a finite group. In this case $\beta\_1^{(2)}(G) = 0$ (since the ordinary homology of $G$ with complex coefficients vanishes). So assume $G$ is infinite.


How are the $L^2$-Betti numbers defined? We need to take a classifying space K(G,1) for $G$. Take a presentation $\langle g\_1,\dots,g\_d | R \rangle$ with some relations $R$. The classifying space $X$ is constructed from a bouquet of $d$ circles by glueing in some higher dimensional cells (which we can ignore here). Passing to the universal covering of $X$ we see that we can compute the $L^2$-Betti number from a complex of Hilbert $\mathcal{N}(G)$-modules:
$$ \dots \stackrel{\partial\_2}{\longrightarrow} L^2(G)^d \stackrel{\partial\_1}{\longrightarrow} L^2(G) \longrightarrow 0$$


The first $L^2$-Betti number is defined as the von Neumann dimension $\beta\_1^{(2)}(G) = \dim\_{\mathcal{N}(G)} \ker(\partial\_1) - \dim\_{\mathcal{N}(G)} \overline{\mathrm{Im}(\partial\_2)}$.
Using the additivity of von Neumann dimension in short weakly exact sequences (Thm. 1.12 in [Lück]) we obtain
$$\beta\_1^{(2)} = \dim\_{\mathcal{N}(G)}(L^2(G)^d) - \dim\_{\mathcal{N}(G)} \overline{\mathrm{Im}(\partial\_1)} - \dim\_{\mathcal{N}(G)} \overline{\mathrm{Im}(\partial\_2)} \leq d - 1 $$
since for an infinite group the map $\partial\_1$ has dense image (Thm.1.35 (8) in [Lück]).


This is just a special case of the Morse inequalities (if $G$ is infinite): Thm 1.35 (7) in [Lück].


[Lück] Lück - $L^2$-Invariants: Theory and Applications to Geometry and K-Theory, Springer.


