On the fields generated by the lengths of closed geodesics in locally symmetric spaces
type: Item
instance of: scholarly article
title: On the fields generated by the lengths of closed geodesics in locally symmetric spaces
zbMATH Open document ID: 1308.53077
author: Gopal Prasad
author: Andrei Rapinchuk
published in: Geometriae Dedicata
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.0141
review text: 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 exactl…
reviewed by: Balasubramanian Sury
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 20G30
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 6358000
zbMATH Keywords: length commensurability
zbMATH Keywords: weak containment
zbMATH Keywords: locally symmetric space
zbMATH Keywords: length of closed geodesic
zbMATH Keywords: trace field
zbMATH Keywords: Schanuel conjecture
MaRDI profile type: Publication
OpenAlex ID: W2100831801
arXiv ID: 1110.0141
cites work: On Schanuel's conjectures
cites work: Q3998726
cites work: Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds
cites work: Outer automorphisms of algebraic groups and determining groups by their maximal tori.
cites work: Weakly commensurable \(S\)-arithmetic subgroups in almost simple algebraic groups of types B and C.
cites work: Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist
cites work: Q4002684
cites work: Cartan subgroups and lattices in semi-simple groups
cites work: Computation of the metaplectic kernel
cites work: Irreducible tori in semisimple groups
cites work: Existence of irreducible \(\mathbb R\)-regular elements in Zariski-dense subgroups
cites work: Zariski-dense subgroups and transcendental number theory
cites work: Number theoretic techniques in the theory of Lie groups and differential geometry
cites work: Weakly commensurable arithmetic groups and isospectral locally symmetric spaces
cites work: Local-global principles for embedding of fields with involution into simple algebras with involution
cites work: Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces.
cites work: Q4063262
cites work: Isospectrality and commensurability of arithmetic hyperbolic 2- and 3- manifolds
cites work: Q3708857
cites work: Linear algebraic groups.
cites work: Q5648549
cites work: RINGS OF DEFINITION OF DENSE SUBGROUPS OF SEMISIMPLE LINEAR GROUPS
cites work: Isospectral Problem of Locally Symmetric Spaces
DOI: 10.1007/S10711-013-9909-3
description: scientific article; zbMATH DE number 6358000