Hyperbolicity vs. Amenability for Planar Graphs

Discrete & Computational Geometry - Tập 58 - Trang 67-79 - 2017
Bruno Federici1, Agelos Georgakopoulos1
1Mathematics Institute, University of Warwick, Coventry, UK

Tóm tắt

The aim of this paper is to clarify the relationship between Gromov-hyperbolicity and amenability for planar maps.

Tài liệu tham khảo

Benjamini, I.: Coarse Geometry and Randomness. Lecture Notes in Mathematics, vol. 2100. Springer, Cham (2013) Bowditch, B.H.: Notes on Gromov’s hyperbolicity criterion for path-metric spaces. In: Ghys, E., Haefliger, A. (eds.) Group Theory from a Geometrical Viewpoint, pp. 64–167. World Scientific, River Edge (1991) Bowditch, B.H.: A short proof that a subquadratic isoperimetric inequality implies a linear one. Mich. Math. J. 42(1), 103–107 (1995) Carmesin, J., Georgakopoulos, A.: Every planar graph with the Liouville property is amenable. arXiv:1502.02542 (2015) Druţu, C., Kapovich, M.: Lectures on Geometric Group Theory. http://people.maths.ox.ac.uk/drutu/tcc2/ChaptersBook (2013) Georgakopoulos, A.: The boundary of a square tiling of a graph coincides with the Poisson boundary. Invent. Math. 203(3), 773–821 (2016) Gromov, M.: Hyperbolic Groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987) Kanai, M.: Rough isometries, and combinatorial approximations of geometries of non-compact Riemannian manifolds. J. Math. Soc. Jpn. 37(3), 391–413 (1985) Northshield, S.: Circle boundaries of planar graphs. Potential Anal. 2(4), 299–314 (1993)