  Let $\Omega$ be a Lipschitz bounded domain of $\mathbb{R}^N $, $N\geq2$. The
fractional Cheeger constant $h_s (\Omega)$, $0<s<1$, is defined by
\[h_s(\Omega)=\inf_{E\subset{\Omega}}\frac{P_s(E)}{|E|},\: \text{ where } \:
P_s (E)=\int_{\mathbb{R}^N }\int_{\mathbb{R}^N
}\frac{|\chi_{E}(x)-\chi_{E}(y)|}{|x-y|^{N+s}} dx dy,\] with $\chi_{E}$
denoting the characteristic function of the smooth subdomain $E$. The main
purpose of this paper is to show that
\[\lim_{p\rightarrow1^+}\left|\phi_p^s\right|_{L^{\infty}(\Omega)}^{1-p}=h_s
(\Omega)=\lim_{p\rightarrow1^+}\left|\phi_p^s\right|_{L^1(\Omega)}^{1-p},\]
where $\phi_p^s$ is the fractional $(s,p)$-torsion function of $\Omega$, that
is, the solution of the Dirichlet problem for the fractional $p$-Laplacian:
$-(\Delta)_p^s\,u=1$ in $\Omega$, $u=0$ in $\mathbb{R}^N \setminus\Omega$. For
this, we derive suitable bounds for the first eigenvalue
$\lambda_{1,p}^s(\Omega)$ of the fractional $p$-Laplacian operator in terms of
$\phi_p^s$. We also show that $\phi_p^s$ minimizes the $(s,p)$-Gagliardo
seminorm in $\mathbb{R}^N $, among the functions normalized by the $L^1$-norm.
