On the fields generated by the lengths of closed geodesics in locally symmetric spaces
The authors continue their study of the length spectra of Riemannian manifolds. In the present paper, they study the field \(F(M)\) generated by the set \(L(M)\) of lengths of closed geodesics for a locally symmetric space \(M\). They show that if \(M_1\), \(M_2\) are two arithmetically defined locally symmetric spaces such that the set \(\mathbb{Q}\cdot L(M_1) \neq \mathbb{Q}\cdot L(M_2)\) (that is, if \(M_1\), \(M_2\) are \textit{not} length-commensurable), then the sets \(L(M_1)\), \(L(M_2)\) are very different from each other. More precisely, under the above hypothesis, the composite field \(F(M_1)F(M_2)\) has infinite transcendence degree over one of these fields. The departure from the earlier work of the authors is a consideration of a new notion called ``weak containment'' -- not to be confused with weak containment of representations. For semisimple groups \(G_1\), \(G_2\) over a field \(F\), a finite set \(S_1\) of semisimple elements of a Zariski-dense subgroup \(\Gamma_1\) of \(G_1(F)\) is said to be weakly contained in a Zariski-dense subgroup \(\Gamma_2\) of \(G_2(F)\) if there is a corresponding finite set \(S_2\) of semisimple elements in \(\Gamma_2\) and there are maximal \(F_i\)-tori containing the elements of \(S_1 \cup S_2\) in the corresponding \(G_i\)'s, and characters on these tori such that the products of the values of these characters at the elements of \(S_1\) match with the corresponding product over \(S_2\) (both being nontrivial). This generalizes the notion of weak commensurability that the authors had considered earlier. The authors obtain conditions for weak containment and use them to show (assuming the truth of Schanuel's conjecture) the infinity of the transcendence degree. Interestingly, as the authors point out, it can happen that \(M_1\), \(M_2\) may not be length commensurable but still it can happen that \(L(M_1) \subset L(M_2)\). For example, if \(D_1\), \(D_2\) are quaternion algebras over \(\mathbb{Q}\) ramifying at exactly \(\{2,3\}\) and \(\{2,3,5,7\}\), consider torsion-free discrete subgroups \(\Gamma_i \leq G_i := \mathrm{PSL}_{1,D_i}\). Then \(\Gamma_1\), \(\Gamma_2\) are both arithmetic subgroups of \(G_1(\mathbb{R}) \cong G_2(\mathbb{R}) \cong\mathrm{PSL}_2(\mathbb{R})\). The corresponding locally symmetric spaces \(M_1 := \Gamma_1 \backslash \mathcal{H}\) and \(M_2 := \Gamma_2 \backslash\mathcal{H}\) are not commensurable. However, since each maximal subfield of \(D_2\) is also a subfield of \(D_1\), the field \(F(M_2) \subset F(M_1)\). Of course, by the authors' main theorem, \(F(M_1)\) has infinite transcendence degree over \(F(M_2)\). It should be stressed that the authors need to assume the truth of Schanuel's conjecture for deriving their results. For general locally symmetric spaces they prove: Let \(G_1\), \(G_2\) be connected, absolutely almost simple \(\mathbb{R}\)-isotropic groups of adjoint type. Let \(\Gamma_i \leq G_i(\mathbb{R})\) be discrete, finitely generated, torsion-free Zariski dense subgroups and \(M_i\) be the corresponding locally symmetric spaces. Denote by \(F_i\), the subfield of \(\mathbb{R}\) generated by the set \(L(M_i)\) of lengths of closed geodesics on \(M_i\). Let \(K_i\) denote the trace field of \(\Gamma_i\) (generated by \(\mathrm{tr}(\mathrm{Ad}_{\gamma})\) as \(\gamma\) varies in \(\Gamma_i\)). Denote by \(w_i\), the order of the absolute Weyl group of \(G_i\). If, either \(w_1>w_2\), or \(w_1=w_2\) but \(K_1 \not\subset K_2\), then \(F_1F_2\) has infinite transcendence degree over \(F_2\). When \(w_1=w_2\), the groups \(G_1\), \(G_2\) have the same Cartan-Killing type (or one of them is \(B_n\) and the other \(C_n\)). When the groups are of the same type, the authors prove a stronger result when \(\Gamma_i\) are arithmetic subgroups. They prove that if they are not length commensurable, the length spectra satisfy not only that \(F_1F_2\) has infinite transcendence degree over one of the two \(F_i\)'s (say, over \(F_1\)) but also satisfy the stronger property that \[ L(M_1) \not\subset A\cdot \mathbb{Q}\cdot L(M_2) \] for any finite set \(A\) of real numbers. In the case of types \(A_n D_{2n+1} (n>1)\), and \(E_6\), they also need to make the assumption that at least one of \(K_1\), \(K_2\) is different from \(\mathbb{Q}\). There are several other interesting results and remarks scattered over the paper. On the way to proving the main results, the authors raise the following question of independent interest: Is it possible to produce number fields \(K_1 \subset K_2\) and construct a \(K_1\)-isotropic simple algebraic group \(G_1\) and a \(K_2\)-anisotropic simple \(G_2\) such that every \(K_1\)-anisotropic torus of \(G_1\) is \(K_2\)-isomorphic to a \(K_2\)-torus of \(G_2\)?
scientific article; zbMATH DE number 6358000
scientific article