arXiv subjects

Kamil Orzechowski

Publications and source records attributed to Kamil Orzechowski.

7 recordsLinked to original sources

arXiv2026

The Banach-Tarski paradox in complete discretely valued fields

We prove some results related to the classical Banach--Tarski paradox in the setting of a field $\mathbb{K}$ that is complete with respect to a discrete non-Archimedean valuation (e.g., when $\mathbb{K}$ is the field $\mathbb{Q}_p$ of $p$-adic numbers for a prime $p$). Namely, the field $\mathbb{K}$, as well as all balls and spheres in $\mathbb{K}$, admit a paradoxical decomposition with respect to the isometry group of $\mathbb{K}$. Such decompositions can be realized using pieces with the Baire property if $\mathbb{K}$ is separable. Under the additional assumption of local compactness of $\mathbb{K}$ (e.g., when $\mathbb{K}=\mathbb{Q}_p$), any two bounded subsets of $\mathbb{K}$ with nonempty interiors are equidecomposable with respect to the isometry group of $\mathbb{K}$. Our results complete the study of paradoxical decompositions in the non-Archimedean setting, addressing the one-dimensional case and building on earlier work for higher-dimensional normed spaces over $\mathbb{K}$ with respect to groups of affine isometries.

math.FA↗

arXiv2025

Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

We show that any normed space $(K^n,\|\cdot\|)$, $n\ge 2$, over a field $K$ equipped with a nontrivial non-Archimedean valuation admits a paradoxical decomposition using four pieces with respect to the group of its affine isometries, provided that the norm $\|\cdot\|$ is equivalent to the maximum norm. It follows that any finite-dimensional normed space $(X,\|\cdot\|)$ with $\dim{X}\ge 2$ over a complete non-Archimedean nontrivially valued field $(K,|\cdot|)$ is paradoxical using four pieces with respect to the group of its affine isometries.

math.FA↗

arXiv2025

Translation length formula for two-generated groups acting on trees

We investigate translation length functions for two-generated groups acting by isometries on $\Lambda$-trees, where $\Lambda$ is a totally ordered abelian group. In this context, we provide an explicit formula for the translation length of any element of the group, under certain assumptions on the translation lengths of its generators and their products. Our approach is purely combinatorial and uses only the defining axioms of pseudo-lengths. As shown by Parry, pseudo-lengths coincide with the translation length functions for actions on $\Lambda$-trees. Furthermore, we prove that, under certain conditions on four elements $\alpha, \beta, \gamma, \delta \in \Lambda$, there exists a unique pseudo-length on the free group $F(a,b)$ assigning these values to $a$, $b$, $ab$, $ab^{-1}$, respectively. Applications include results on properly discontinuous actions and discrete free groups of isometries. We also develop an algorithmic approach to studying translation length functions arising from free actions on $\mathbb{R}$-trees. Based on this, we state a conjecture that would lead to a description of $\mathrm{Aut}{(F_2)}$-orbits in the Culler-Vogtmann outer space.

math.GR↗

arXiv2024

The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields

We show some results related to the classical Banach-Tarski paradox in the setting of finite-dimensional normed spaces over a non-Archimedean valued field $K$. For instance, all balls and spheres in $K^n$, and the whole space $K^n$ (for $n\ge 2$) are paradoxical with respect to certain groups of isometries of $K^n$. If $K$ is locally compact (e.g., $K$ is the field $\mathbb{Q}_p$ of $p$-adic numbers for any prime number $p$), any two bounded subsets of $K^n$ with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of $K^n$ (for $n\ge 2$).

math.FA↗

arXiv2019

APD profiles and transfinite asymptotic dimension

We develop the theory of APD profiles introduced by J. Dydak for $\infty$-pseudometric spaces. We connect them with transfinite asymptotic dimension defined by T. Radul. We give a characterization of spaces with transfinite asymptotic dimension at most $ω+n$ for $n\inω$ and a sufficient condition for a space to have transfinite asymptotic dimension at most $m\cdot ω+n$ for $m,n\inω$, using the language of APD profiles.

math.MG↗

arXiv2019

Asymptotic property C of the countable direct sum of uniformly discrete $0$-hyperbolic spaces

We define the direct sum of a countable family of pointed metric spaces in a way resembling the direct sum of groups. Then we prove that if a family consists of $0$-hyperbolic (in the sense of Gromov) and $D$-discrete spaces, then its direct sum has asymptotic property C. The main example is a countable direct sum of free groups of (possibly varying) finite rank. This is a generalization of T. Yamauchi's result concernig the countable direct sum of the integers.

math.GN↗

arXiv2016

Characterization of the Haagerup property for residually amenable groups

The notions of a box family and fibred cofinitely-coarse embedding are introduced. The countable, residually amenable groups satisfying the Haagerup property are then characterized as those possessing a box family that admits a fibred cofinitely-coarse embedding into a Hilbert space. This is a generalization of a result of X. Chen, Q. Wang and X. Wang on residually finite groups.

math.GR↗