Tag - Dehn function
The Dehn function was introduced by computer scientists Madlener and Otto to describe the complexity of the word problem of a group, and also by Gromov as a geometric invariant of finitely presented groups. In this talk, I will show that the upper bound of the Dehn function of finitely presented metabelian group G is 2n2k, where k is the torsion-free rank of the abelianization Gab, answering the question that if the Dehn functions of metabelian groups are uniformly bounded. I will also talk about the relative Dehn function of finitely generated metabelian group and its relation to the Dehn function.
Bridson, Burillo, Elder and Šunić asked if there exists a group with intermediate geodesic growth and if there is a characterization of groups with polynomial geodesic growth. Towards these questions, they showed that there is no nilpotent group with intermediate geodesic growth, and they provided a sufficient condition for a virtually abelian group to have polynomial geodesic growth. In this talk, we take the next step in this study and show that the geodesic growth for a finitely generated virtually abelian group is either polynomial or exponential; and that the generating function of this geodesic growth series is holonomic, and rational in the polynomial growth case. To obtain this result, we will make use of the combinatorial properties of the class of linearly constrained language as studied by Massazza. In addition, we show that the language of geodesics of a virtually abelian group is blind multicounter.
We will survey new and old results on the "energy'' of a set of affine transformations and see applications in the geometric side of additive combinatorics, in combinatorial geometry and in questions on growth in the affine group. In particular we will answer to the affirmative a question of Yufei Zhao.
The Dehn function of a finitely presented group, first introduced by Gromov, is a useful invariant that is closely related to the solvability of the group’s word problem. It is well-known that a finitely presented group is word hyperbolic if and only if it has sub-quadratic (and thus linear) Dehn function. A result of Ghys and de la Harpe states that no word hyperbolic group can have a (finitely generated) infinite torsion subgroup. We show that the same does not hold for finitely presented groups with Dehn function as small as cubic. In particular, for every m≥2 and sufficiently large odd integer n, there exists an embedding of the free Burnside group B(m,n) into a finitely presented group with cubic Dehn function.
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