Journal ArticleDOI
Connection blocking in $$\text {SL}(n,\mathbb {R})$$ SL ( n , R ) quotients
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TLDR
In this paper, the authors show that the obtained quotient homogeneous spaces are not finitely blockable and that the set of non-blackable pairs is a dense subset of the homogeneous space M 2 times M 2.Abstract:
Let G be a connected Lie group and
$$\varGamma \subset G$$
a lattice. Connection curves of the homogeneous space
$$M=G/\varGamma $$
are the orbits of one parameter subgroups of G. To block a pair of points
$$m_1,m_2 \in M$$
is to find a finite set
$$B \subset M{\setminus } \{m_1, m_2 \}$$
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
$$M_n= \text {SL}(n,\mathbb {R})/\varGamma $$
, where
$$\varGamma =\text {SL}(n,\mathbb {Z})$$
is the integer lattice. We focus on
$$M_2$$
and show that the set of non blackable pairs is a dense subset of
$$M_2 \times M_2$$
, and we conclude manifolds
$$M_n$$
are not blockable. Finally, we review a quaternionic structure of
$$\text {SL}(2,\mathbb {R})$$
and a way for making co-compact lattices in this context. We show that the obtained quotient homogeneous spaces are not finitely blockable.read more
References
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BookDOI
Discrete subgroups of semisimple Lie groups
TL;DR: The Structure of the Book as discussed by the authors is a collection of essays about algebraic groups over arbitrary fields, including a discussion of the relation between the structure of closed subgroups and property (T) of normal subgroups.
Book
Introduction to Arithmetic Groups
TL;DR: A good introduction to the study of arithmetic subgroups of semisimple Lie groups can be found in this article, where the authors provide primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, and quasi-isometries.
Book
Lie Groups: An Introduction through Linear Groups
TL;DR: Inverse functions and Inverse function theorem Theorem Theorem 1.1 as discussed by the authors The exponential map is a special case of Inverse Function Theorem 2.2.1.
Journal ArticleDOI
Growth of the number of geodesics between points and insecurity for Riemannian manifolds
Keith Burns,Eugene Gutkin +1 more
TL;DR: Gromov and Mane as discussed by the authors proved that the number of geodesics with length between every pair of points in a uniformly secure Riemannian manifold grows polynomially as $T \to \infty.
Journal ArticleDOI
Blocking of billiard orbits and security for polygons and flat surfaces
E. Gutkin,E. Gutkin +1 more
TL;DR: In this article, the notion of security for polygons and flat surfaces was introduced and studied, and it was shown that a lattice polygon is secure iff it is arithmetic.