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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?

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    $\begingroup$ What is $\beta^2(G)$? $\endgroup$ – Andy Putman Nov 3 '16 at 2:35
  • $\begingroup$ @AndyPutman the first $l^2$-Betti number of $G$. $\endgroup$ – Katherine Nov 3 '16 at 2:40
  • $\begingroup$ The key is the Morse inequality for $\ell^2$-betti numbers; see Corollary 3.6.3 of homepages.warwick.ac.uk/~masmbh/files/… $\endgroup$ – Andy Putman Nov 3 '16 at 3:02
  • $\begingroup$ @AndyPutman thank you, but can you offer more information about how to apply the corollary to the inequality? $\endgroup$ – Katherine Nov 3 '16 at 3:20
  • $\begingroup$ Katherine, regarding Andy's answer you should consider $X$ to be standard $K(G,1)$ space having one 0-cell and $d(G)$ many 1-cells and take $k=1$ in the Morse inequality. As for a specific reference, did you try Lueck's book? $\endgroup$ – Uri Bader Nov 3 '16 at 16:58
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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.

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