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Undirected and directed graphs with near polynomial growth

V.I. Trofimov (2003)

Discussiones Mathematicae Graph Theory

The growth function of a graph with respect to a vertex is near polynomial if there exists a polynomial bounding it above for infinitely many positive integers. In the paper vertex-symmetric undirected graphs and vertex-symmetric directed graphs with coinciding in- and out-degrees are described in the case their growth functions are near polynomial.

Uniform growth of groups acting on Cartan–Hadamard spaces

Gérard Besson, Gilles Courtois, Sylvestre Gallot (2011)

Journal of the European Mathematical Society

In this paper we investigate the growth of finitely generated groups. We recall the definition of the algebraic entropy of a group and show that if the group is acting as a discrete subgroup of the isometry group of a Cartan–Hadamard manifold with pinched negative curvature then a Tits alternative is true. More precisely the group is either virtually nilpotent or has a uniform growth bounded below by an explicit constant.

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