In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis in [19] is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.

In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering [6, Question D] posed by Behrstock, Hagen, and Sisto. In order to operate in such a general setting, we establish a number of structural results for hierarchically hyperbolic spaces and hieromorphisms (that is, morphisms between such spaces), and we introduce two new notions for hierarchical hyperbolicity, that is concreteness and the intersection property, proving that they are satisfied in all known examples.

Let be a class of groups. We give sufficient conditions ensuring that a free product of residually groups is again residually , and analogous conditions are given for LEgroups. As a corollary, we obtain that the class of residually amenable groups and the one of locally embeddable into amenable (LEA) groups are closed under taking free products.Moreover, we consider the pro-topology and we characterize special HNN extensions and amalgamated free products that are residually , where is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.

In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering [6, Question D] posed by Behrstock, Hagen, and Sisto. In order to operate in such a general setting, we establish a number of structural results for hierarchically hyperbolic spaces and hieromorphisms (that is, morphisms between such spaces), and we introduce two new notions for hierarchical hyperbolicity, that is concreteness and the intersection property , proving that they are satisfied in all known examples.

We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products.Furthermore, we develop the tools to study the analogous question in the pro-p case. For a wide class of groups we show that the relevant cyclic subgroups -which are called p-isolated -are closed in the pro-p topology of the graph product. In particular, we show that every pisolated cyclic subgroup of a right-angled Artin group is closed in the pro-p topology, and we fully characterise such subgroups.