THESIS PRE-DEFENSE -- Growth in groups of non-positive curvature

Speaker:  Xabier Legaspi (ICMAT-UCM)
Advisor/s:  Yago Antolín (ICMAT-UCM); Rémi Coulon (CNRS-Université de Rennes 1)
Date:  Wednesday, 07 June 2023 - 10:30
Place:  Aula 115, Facultad de Matemáticas, Universidad Complutense de Madrid


The aim of this thesis is to obtain a better understanding of the behaviour of exponential growth rates within the class of groups that act acylindrically in a hyperbolic space in the sense of Gromov. To do this, we will address two problems of a different nature.

In the first problem we will study the exponential growth rates of quasi-convex subgroups. We will compare these rates with that of the ambient group and we will determine when it is possible to obtain strict equality/inequality. To do so, we will exploit proper actions on metric spaces that, a priori, are not hyperbolic, but that have isometries that behave like the loxodromic isometries of a hyperbolic space.

The second problem revolves around uniform uniform exponential growth. We will prove that this property is preserved if we take small cancellation quotients of groups that act acylindrically on a hyperbolic space. As a corollary, we will obtain that there is a universal lower bound on the uniform exponential growth rate for the family of classical small cancellation quotients. This bound depends only on one of the two acylindricity parameters.

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