Growth of \(k\)-dimensional systoles in congruence coverings (Q6581835)

Let \(M\) be an arithmetic locally symmetric space of dimension \(n\), and let \(\{ M_i\to M\}_{i=1,2,\ldots}\) be a sequence of its regular congruence coverings of degrees \(d_i\). The absolute \(k\)-dimensional systole \(\mathrm{absys}_k(M)\) of \(M\) is defined to be the infimum of \(k\)-dimensional volumes of the subsets of \(M\) which cannot be homotoped to \((k-1)\)-dimensional subsets in \(M\). Then \(\mathrm{absys}_n(M_i)=\mathrm{vol}(M_i)=\mathrm{vol}(M)d_i\). It was shown by \textit{L. Guth} and \textit{A. Lubotzky} [J. Math. Phys. 55, No. 8, 082202, 13 p. (2014; Zbl 1298.81052)] that \(\mathrm{absys}_1(M_i)\) has the same degree of growth as \(c_1\log(d_i)\), for a positive constant \(c_1\). The authors are interested in the growth rate of the intermediate systoles \(\mathrm{absys}_k(M_i)\), \(k=2,\ldots, n-1\), in such sequences.\N\NThe question about the growth of \(k\)-dimensional systoles in congruence coverings was first raised by \textit{M. Gromov} [Sémin. Congr. 1, 291--362 (1996; Zbl 0877.53002)]]. In particular, he indicated that one might expect to have polylogarithmic (i.e., polynomial in the logarithm) growth of \(\mathrm{absys}_k(M_i)\), for \(k\leq {\mathbb{R}}\)-\(\mathrm{rank}(M)\) and a constant power growth for bigger \(k\). Assume \(M\) is a compact arithmetic locally symmetric space of dimension \(n\) and of real rank \(r\), and assume \(\{ M_i\to M\}\) is a sequence of congruence coverings of degrees \(d_i\). Let \(r_1=r_1(M)\leq r\) be a parameter of the root system associated to the algebraic group defining \(M\). The main result of the paper says that \(\mathrm{absys}_k(M_i)\) grows polylogarithmically with \(d_i\) for \(1\leq k\leq \max\{ 1, r_1\}\) and as a power function for \(r<k\leq n\).\N\NThe authors expect that between the ranks \(r_1(M)\) and \(r(M)\) the polylogarithmic growth may not persist but that the constant power growth also occurs in this range. The existence of this intermediate range is a new phenomenon, which can only be observed for symmetric spaces of certain types. For example, the spaces corresponding to higher rank simple Lie groups of type A always have this property while the spaces corresponding to simple split groups of type B do not have it. One of the main goals of the authors is to point out the possibility of a variety of behaviours in the intermediate range.\N\NThe authors also consider homological systoles of the congruence coverings. Here the \(k\)-dimensional systole \(\mathrm{sys}_k(M;A)\) is defined as the infimum of the \(k\)-dimensional volumes of the \(k\)-cycles in \(M\) with coefficients in \(A\) which are not homologous to zero in \(M\). They give bounds for homological systoles of the congruence coverings with the coefficients in \({\mathbb{Q}}\), \({\mathbb{Z}}\) and \({\mathbb{Z}}/p{\mathbb{Z}}\).