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1. Approximation of $L_2$-torsion.

    1.     Let $M$ be a closed Riemannian manifold and let $$\Gamma=\pi_1(M)>\Gamma_1>\Gamma_2>\ldots$$ be a decreasing sequence of finite index normal subgroups of $\Gamma$ with $\bigcap\Gamma_n=1$. Let $\tilde{M}$ be the universal cover of $M$ and let $M_n=\tilde{M}/\Gamma_n$.

      In addition, suppose that $M$ is aspherical with $\dim M=2k+1$ and $\beta_j^{(2)}=0$ for every $j$.

      Add remark

      Problem 1.1.

      [Wolfgang Lück] Do we have $$\rho^{(2)}(\tilde{M})=(-1)^k\lim \frac{\log(|tors (H_k(M_n))|)}{|\Gamma:\Gamma_n|}?$$
        1. Remark. Note that, if $M$ is a hyperbolic $3$-manifold then $\rho^{(2)}(\tilde{M})=-\frac{1}{6\pi}vol(M)$. Moreover, if $\Gamma$ has an elementary amenable normal subgroup, then Question holds for $\Gamma$.
            • Remark. Note that, the question is true for groups that are limits of left-orderable amenable groups. Also for groups where every non-trivial finitely generated subgroup surjects on $\Z$).

                •     Let $M$ be a closed Riemannian manifold and let $$\Gamma=\pi_1(M)>\Gamma_1>\Gamma_2>\ldots$$ be a decreasing sequence of finite index normal subgroups of $\Gamma$ with $\bigcap\Gamma_n=1$. Let $\tilde{M}$ be the universal cover of $M$ and let $M_n=\tilde{M}/\Gamma_n$.

                  Add remark

                  Problem 1.2.

                  [Wolfgang Lück] Is $$\rho^{(2)}(\tilde{M})=\lim\limits_n \frac{\rho(M_n)}{|\Gamma:\Gamma_n|}?$$
                      Here $\rho$ is the Ray-Singer torsion and $\rho^{(2)}$ is $L_2$-version of it.

                    •     The following is equivalent to the previous problem.

                      Add remark

                      Problem 1.3.

                      Let $A\in M_l(\mathbb{Z} \Gamma)$, denote $A_k=A/\Gamma_k$. Is it true that $$tr_{L\Gamma}(\log A^*A)=\lim \frac{tr(\log A_k^*A_k)}{|\Gamma:\Gamma_k|}?$$
                          Here $tr_{L\Gamma}$ is the canonical trace on the group von Neumann algebra $L\Gamma$.


                        • Add remark

                          Problem 1.4.

                          Is $$\lim \frac{b_{F_p}(\Gamma_n)}{|\Gamma : \Gamma_n|}=\lim \frac{rk(\Gamma_n)}{|\Gamma:\Gamma_n|}$$ for any (not necessarily finitely generated) group?

                            •     Let $\Gamma$ be a finitely presented residually $p$-group. Let $\Gamma_n$ be a normal $p$-chain with $\bigcap \Gamma_n=1$, then $$\lim \frac{b_{\mathbb{Q}}(\Gamma_n)}{|\Gamma:\Gamma_n|}\leq \lim \frac{b_{F_p}}{|\Gamma: \Gamma_n|}\leq \lim \frac{rk(\Gamma_n)}{|\Gamma:\Gamma_n|}$$

                              Add remark

                              Problem 1.5.

                              [Misha Ershov] Can these inequalities be strict?

                                  Cite this as: AimPL: $L^2$ invariants for finitely generated groups, available at http://aimpl.org/l2invariantsgroups.