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Karol Duda

Publications and source records attributed to Karol Duda.

9 recordsLinked to original sources
arXiv2026

Local-to-global fixed point properties for graphical C(4)-T(4) and C(6) small cancellation complexes

Graphical small cancellation extends the classical small cancellation theory and provides a powerful method for constructing groups with prescribed subgraphs in their Cayley graphs. We prove that torsion subgroups of groups defined by possibly infinite C(4)-T(4) graphical small cancellation presentations are finite. We also prove the corresponding result for groups defined by C(6) graphical small cancellation presentations under the additional assumption that the presentation is torsion-essentially C(6)-free. Both results follow from a more general result on local-to-global fixed point properties for torsion groups acting by automorphisms on simply connected graphical small cancellation complexes.

math.GR↗
arXiv2026

The small cancellation flat torus theorem

We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of $\mathbb{Z}^2$ without invariant flat, and hence not admitting any CAT(0) metric invariant under automorpihsms. We introduce the notion of thickened-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro.

math.GR↗
arXiv2025

Hall's Harem Theorem with controlled sizes of cycles

We prove a new version of Hall's Harem Theorem, where the final matching is realized by a unary function with additional conditions on behavior of cycles. The present paper can be considered as a helpful companion of the paper of the author: arXiv:2105.06304, where a computable version of Hall's Harem Theorem with controlled sizes of cycles is proved. These two versions of Hall's Harem Theorem are independent: none of them follows from the other one.

math.CO↗
arXiv2025

Computable Folner sequences of amenable groups

The paper considers computable Folner sequences in computably enumerable amenable groups. We extend some basic results of M. Cavaleri on existence of such sequences to the case of groups where finite generation is not assumed. We also initiate some new directions in this topic, for example complexity of families of effective Folner sequences. Possible extensions of this approach to metric groups are also discussed. This paper also contains some unpublished results from the paper of the first author arXiv:1904.02640.

math.GR↗
arXiv2023

The Cohen--Lyndon property in non-metric small-cancellation

We show that the Cohen--Lyndon property holds for all non-metric small-cancellation quotients. This generalises the analogous result from the metric small-cancellation setting, and answers a question asked by Lyndon in 1966 and by Wall in his 1979 problem list.

math.GR↗
arXiv2022

Amenability and computability

In this paper we extend the approach of M. Cavaleri to effective amenability to the class of computably enumerable groups, i.e. in particular we do not assume that groups are finitely generated. In the case of computable groups we also study complexity of the set of all effective Følner sequences and effective paradoxical decomposition. In the appendix we attach a version of the paper "On decidability of amenability in computable groups" by K. Duda and A. Ivanov which covers Sections 9 and 10 from the previous version.

math.GR↗
arXiv2021

Torsion subgroups of small cancellation groups

We prove that torsion subgroups of groups defined by C(6), C(4)-T(4) or C(3)-T(6) small cancellation presentations are finite cyclic groups. This follows from a more general result on the existence of fixed points for locally elliptic (every element fixes a point) actions of groups on simply connected small cancellation complexes. We present an application concerning automatic continuity. We observe that simply connected C(3)-T(6) complexes may be equipped with a CAT(0) metric. This allows us to get stronger results on locally elliptic actions in that case. It also implies that the Tits Alternative holds for groups acting on simply connected C(3)-T(6) small cancellation complexes with a bound on the order of cell stabilisers.

math.GR↗