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On the finite blocking property

Thierry Monteil (2005)

Annales de l’institut Fourier

A planar polygonal billiard 𝒫 is said to have the finite blocking property if for every pair ( O , A ) of points in 𝒫 there exists a finite number of “blocking” points B 1 , ⋯ , B n such that every billiard trajectory from O to A meets one of the B i ’s. Generalizing our construction of a counter-example to a theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the finite blocking property are the square, the equilateral triangle and the hexagon. Then we extend this result to translation surfaces....

On the Haagerup inequality and groups acting on A ˜ n -buildings

Alain Valette (1997)

Annales de l'institut Fourier

Let Γ be a group endowed with a length function L , and let E be a linear subspace of C Γ . We say that E satisfies the Haagerup inequality if there exists constants C , s > 0 such that, for any f ∈ E , the convolutor norm of f on ℓ 2 ( Γ ) is dominated by C times the ℓ 2 norm of f ( 1 + L ) s . We show that, for E = C Γ , the Haagerup inequality can be expressed in terms of decay of random walks associated with finitely supported symmetric probability measures on Γ . If L is a word length function on a finitely generated group Γ , we show that,...

On the Infinite Loch Ness monster

John A. Arredondo, Camilo Ramírez Maluendas (2017)

Commentationes Mathematicae Universitatis Carolinae

In this paper we introduce the topological surface called {Infinite Loch Ness monster}, discussing how this name has evolved and how it has been historically understood. We give two constructions of this surface, one of them having translation structure and the other hyperbolic structure.

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