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We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is arbitrarily large. This suggests that the existing methods for finding non-free groups with rational parabolic generators may be inadequate in this case, as they depend on the presence of relators with few syllables. Our results rely on two variants of the ping-pong lemma that we develop, applicable to groups that are possibly non-free. These variants aim to isolate the group elements responsible for the failure of the classical ping-pong lemma.

We find sufficient conditions for the singularity of a substitution Z -action’s spectrum, which generalize a result of Bufetov and Solomyak, and we also obtain a similar statement for a collection of substitution R -actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector v ∈ Q d , we find necessary and sufficient conditions for the orbit of ω v to be uniformly distributed modulo 1 for almost every ω ∈ R . We use our results to find new examples of singular substitution Z - and R -actions.

We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is arbitrarily large. This suggests that the existing methods for finding non-free groups with rational parabolic generators may be inadequate in this case, as they depend on the presence of relators with few syllables. Our results rely on two variants of the ping-pong lemma that we develop, applicable to groups that are possibly non-free. These variants aim to isolate the group elements responsible for the failure of the classical ping-pong lemma.

We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space of all invariant probability measures, and that if the system is ergodic with respect to Haar measure, then the finitely supported ergodic invariant measures are also dense. A key ingredient in the proof is a variant of the specification property, which we establish for the ergodic systems in this class. Our results also yield the following two consequences: first, that every finitely generated group that is a semidirect product of \\(\\mathbb{Z}\\) with a countable abelian group \\(G\\) is Hilbert-Schmidt stable; and second, a Livshitz-type theorem characterizing the uniform closure of coboundaries arising from continuous functions in terms of vanishing on periodic orbits. We also construct an example showing that, in general dynamical systems, the property of having dense finitely supported invariant measures does not pass to product systems.

We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is arbitrarily large. This suggests that the existing methods for finding non-free groups with rational parabolic generators may be inadequate in this case, as they depend on the presence of relators with few syllables. Our results rely on two variants of the ping-pong lemma that we develop, applicable to groups that are possibly non-free. These variants aim to isolate the group elements responsible for the failure of the classical ping-pong lemma.

We find sufficient conditions for the singularity of a substitution \\(\\mathbb{Z}\\)-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution \\(\\mathbb{R}\\)-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector \\(\\mathbf{v}\\in\\mathbb{Q}^d\\), we find necessary and sufficient conditions for the orbit of \\(\\omega\\mathbf{v}\\) to be uniformly distributed modulo \\(1\\) for almost every \\(\\omega\\in\\mathbb{R}\\). We use our results to find new examples of singular substitution \\(\\mathbb{Z}\\)- and \\(\\mathbb{R}\\)-actions.