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Jacek Świątkowski

Publications and source records attributed to Jacek Świątkowski.

6 recordsLinked to original sources
arXiv2026

$E\mathcal{Z}$-boundaries, splittings over finite subgroups, and dense amalgams

The dense amalgam is an operation (introduced in arXiv:1410.4989) which to any finite collection of metrizable compacta associates canonically some new highly disconnected compact metrisable space in which embedded copies of the initial spaces are appropriately uniformly and disjointly distributed. We show that in the very general framework of $E\mathcal{Z}$-boundaries (unifying many frameworks such as Gromov boundaries, CAT(0)-boundaries, systolic boundaries, etc.), any boundary $Z$ of an infinitely ended group $Γ$ equipped with an appropriate splitting along finite subgroups has a form of the dense amalgam of the limit sets in $Z$ of the factor subgroups of this splitting.

math.GR↗
arXiv2025

Trees of Graphs as Boundaries of Hyperbolic Groups

We characterize those 1-ended word hyperbolic groups whose Gromov boundaries are homeomorphic to trees of graphs (i.e. to inverse limits of graphs that have particularly simple finitary descriptions). These are groups with the simplest connected Gromov boundaries of topological dimension 1. The characterization is expressed in terms of algebraic properties of the Bowditch JSJ splitting of the corresponding groups (i.e. the canonical JSJ splitting over 2-ended subgroups).

math.GR↗
arXiv2019

Reflection trees of graphs as boundaries of Coxeter groups

To any finite graph $X$ (viewed as a topological space) we assosiate some explicit compact metric space ${\cal X}^r(X)$ which we call {\it the reflection tree of graphs $X$}. This space is of topological dimension $\le1$ and its connected components are locally connected. We show that if $X$ is appropriately triangulated (as a simplicial graph $Γ$ for which $X$ is the geometric realization) then the visual boundary $\partial_\infty(W,S)$ of the right angled Coxeter system $(W,S)$ with the nerve isomorphic to $Γ$ is homeomorphic to ${\cal X}^r(X)$. For each $X$, this yields in particular many word hyperbolic groups with Gromov boundary homeomorphic to the space ${\cal X}^r(X)$.

math.GR↗
arXiv2016

Right-angled Coxeter groups with n-dimensional Sierpiński compacta as boundaries

For arbitrary integer n, we describe a large class of right-angled Coxeter systems for which the visual baundary (of the corresponding Coxeter-Davis complex) is homeomorphic to the n-dimensional Sierpiński compactum. We also provide a necessary and sufficient condition for a planar simplicial complex L under which the right-angled Coxeter system whose nerve is L has the visual boundary homeomorphic to the Sierpiński curve.

math.GT↗