Connection blocking in $$\text {SL}(n,\mathbb {R})$$ SL ( n , R ) quotients
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Connection blocking in SL(n, R) quotients Mohammadreza Bidar1 Received: 17 May 2017 / Accepted: 13 March 2020 © Springer Nature B.V. 2020
Abstract Let G be a connected Lie group and Γ ⊂ G a lattice. Connection curves of the homogeneous space M = G/Γ are the orbits of one parameter subgroups of G. To block a pair of points m 1 , m 2 ∈ M is to find a finite set B ⊂ M\{m 1 , m 2 } such that every connecting curve joining m 1 and m 2 intersects B. The homogeneous space M is blockable if every pair of points in M can be blocked. In this paper we investigate blocking properties of Mn = SL(n, R)/Γ , where Γ = SL(n, Z) is the integer lattice. We focus on M2 and show that the set of non blackable pairs is a dense subset of M2 × M2 , and we conclude manifolds Mn are not blockable. Finally, we review a quaternionic structure of SL(2, R) and a way for making co-compact lattices in this context. We show that the obtained quotient homogeneous spaces are not finitely blockable. Keywords Connection blocking · Homogeneous spaces · SL(n, R) · Modified time · Cocompact lattices Mathematics Subject Classification Primary 53C30 · Secondary 57S25
1 Introduction Finite blocking is an interesting concept originating as a problem in billiard dynamics and later studied in the context of Riemannian manifolds. Let (M, g) be a complete, connected, infinitely differentiable Riemannian manifold. For a pair of (not necessarily distinct) points m 1 , m 2 ∈ M let Γ (m 1 , m 2 ) be the set of geodesic segments joining these points. A set B ⊂ M\{m 1 , m 2 } is blocking if every γ ∈ Γ (m 1 , m 2 ) intersects B. The pair m 1 , m 2 is secure if there is a finite blocking set B = B(m 1 , m 2 ). A manifold is secure if all pairs of points are secure. If there is a uniform bound on the cardinalities of blocking sets, the manifold is uniformly secure and the best possible bound is the blocking number. Now, the first question naturally arising is what Riemannian manifolds are secure. If we focus on closed Riemannian manifolds, there is the following conjecture [3,11]: Conjecture 1 A closed Riemannian manifold is secure if and only if it is flat.
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Mohammadreza Bidar [email protected] Department of Mathematics, Albion College, Albion, MI 49224, USA
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Geometriae Dedicata
Flat manifolds are uniformly secure, and the blocking number depends only on their dimension [6,8]. They are also midpoint secure, i.e., the midpoints of connecting geodesics yield a finite blocking set for any pair of points [1,6,8]. Conjecture 1 says that flat manifolds are the only secure manifolds. This has been verified for several special cases: A manifold without conjugate points is uniformly secure if and only if it is flat [3,11]; a compact locally symmetric space is secure if and only if it is flat [8]; the generic manifold is insecure [4,5,9]; Conjecture 1 holds for compact Riemannian surfaces with genus bigger or equal than one [1]; any Riemannian metric has an arbitrarily close, insecure metric in the same conformal class [9]. Gutkin [7] initiated the study of blocking
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