27 citations · 116 across the 31 of their papers we have counts for
43 papers
Elementwise conservative actions and new constructions of boomerang subgroups
Yair Glasner, Tobias Hartnick, Waltraud Lederle
We show that countable non-abelian free groups admit uncountably many mutually singular elementwise conservative non-singular random subgroups, which are supported on infinite subg…
Siegel-Radon transforms of transverse dynamical systems
Michael Björklund, Tobias Hartnick
We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group , to a broader setting in which one of the…
Hurewicz and Dranishnikov-Smith theorems for asymptotic dimension of countable approximate groups
Tobias Hartnick, Vera Tonić
We establish two main results for the asymptotic dimension of countable approximate groups. The first one is a Hurewicz type formula for a global morphism of countable approximate…
Dynamics on spaces of quasimorphisms and applications to approximate lattice theory
Michael Björklund, Tobias Hartnick
We study the dynamics of countable groups on their respective spaces of quasimorphisms. For cohomologically non-trivial quasimorphisms we show that there are no invariant measures…
Automorphism groups of cocompact CAT(0) cube complexes and simplicity
Tobias Hartnick, Merlin Incerti-Medici
We provide a systematic description of the automorphism groups of specially cocompact CAT(0) cube complexes. We show that these groups are topologically finitely generated, present…
Automorphisms of self-similar trees
Tobias Hartnick, Merlin Incerti-Medici
We explicitly determine the automorphism groups of all self-similar trees (a.k.a. trees with finitely many cone types). We show that any such automorphism group is a direct limit o…
A Quillen Stability Criterion for Bounded Cohomology
Carlos De la Cruz Mengual, Tobias Hartnick
We provide a version of Quillen's homological stability criterion for continuous bounded cohomology. This criterion is exploited in the companion paper (arXiv:2201.03879) in order…
Hyperuniformity and non-hyperuniformity of quasicrystals
Michael Björklund, Tobias Hartnick
We develop a general framework to study hyperuniformity of various mathematical models of quasicrystals. Using this framework we provide examples of non-hyperuniform quasicrystals…
Higher finiteness properties of arithmetic approximate lattices: The Rank Theorem for number fields
Tobias Hartnick, Stefan Witzel
We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reduct…
Stabilization of Bounded Cohomology for Classical Groups
Carlos De la Cruz Mengual, Tobias Hartnick
We show that bounded cohomology stabilizes along sequences of classical Lie groups, and along sequences of lattices in them. Our method is based on a criterion from (arXiv:2307.128…
Efficient computations with counting functions on free groups and free monoids
Tobias Hartnick, Alexey Talambutsa
We present efficient algorithms to decide whether two given counting functions on non-abelian free groups or monoids are at bounded distance from each other and to decide whether t…
Symbolic substitution systems beyond abelian groups
Siegfried Beckus, Tobias Hartnick, Felix Pogorzelski
In this article we construct the first examples of strongly aperiodic linearly repetitive Delone sets in non-abelian Lie groups by means of symbolic substitutions. In particular, w…
Intersection spaces and multiple transverse recurrence
Michael Björklund, Tobias Hartnick, Yakov Karasik
We study multiple recurrence properties along separated cross sections for pmp actions of unimodular lcsc group on Polish spaces. We establish a multiple transverse recurrence theo…
Foundations of geometric approximate group theory
Matthew Cordes, Tobias Hartnick, Vera Tonić
We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Björklund and the second-named author. Our theory is based on the notio…
Aperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula
Michael Björklund, Tobias Hartnick, Felix Pogorzelski
We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of…
Linear repetitivity beyond abelian groups
Siegfried Beckus, Tobias Hartnick, Felix Pogorzelski
We show that linearly repetitive weighted Delone sets in groups of polynomial growth have a uniquely ergodic hull. This result applies in particular to the linearly repetitive weig…
Borel density for approximate lattices
Michael Björklund, Tobias Hartnick, Thierry Stulemeijer
We extend classical density theorems of Borel and Dani--Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proo…
Stability in Bounded Cohomology for Classical Groups, I: The Symplectic Case
Carlos De la Cruz Mengual, Tobias Hartnick
We show that continuous bounded group cohomology stabilizes along the sequences of real or complex symplectic Lie groups, and deduce that bounded group cohomology stabilizes along…
Spectral theory of approximate lattices in nilpotent Lie groups
Michael Björklund, Tobias Hartnick
We show that an approximate lattice in a nilpotent Lie group admits a relatively dense subset of central -Bragg peaks for every . For the Heisenberg group we deduce th…
Analytic properties of approximate lattices
Michael Björklund, Tobias Hartnick
We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-…
Aperiodic order and spherical diffraction, II: Translation bounded measures on homogeneous spaces
Michael Björklund, Tobias Hartnick, Felix Pogorzelski
We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we stu…
Kac-Moody symmetric spaces
Walter Freyn, Tobias Hartnick, Max Horn +1
In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-…
Approximate lattices
Michael Björklund, Tobias Hartnick
In this article we introduce and study uniform and non-uniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense…
Bounded cohomology and virtually free hyperbolically embedded subgroups
Tobias Hartnick, Alessandro Sisto
Using a probabilistic argument we show that the second bounded cohomology of an acylindrically hyperbolic group (e.g., a non-elementary hyperbolic or relatively hyperbolic grou…