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Piotr Przytycki

Publications and source records attributed to Piotr Przytycki.

At least 19 recordsLinked to original sources
arXiv2024

The moduli space of cactus flower curves and the virtual cactus group

The space $ \ft_n = \C^n/\C $ of $n$ points on the line modulo translation has a natural compactification $ \overline \ft_n $ as a matroid Schubert variety. In this space, pairwise distances between points can be infinite; it is natural to imagine points at infinite distance from each other as living on different projective lines. We call such a configuration of points a ``flower curve'', since we picture the projective lines joined into a flower. Within $ \ft_n $, we have the space $ F_n = \C^n \setminus Δ/ \C $ of $ n$ distinct points. We introduce a natural compatification $ \overline F_n $ along with a map $ \overline F_n \rightarrow \overline \ft_n $, whose fibres are products of genus 0 Deligne-Mumford spaces. We show that both $\overline \ft_n$ and $\overline F_n$, are special fibers of $1$-parameter families whose generic fibers are, respectively, Losev-Manin and Deligne-Mumford moduli spaces of stable genus $0$ curves with $n+2$ marked points. We find combinatorial models for the real loci $ \overline \ft_n(\BR) $ and $ \overline F_n(\BR) $. Using these models, we prove that these spaces are aspherical and that their equivariant fundamental groups are the virtual symmetric group and the virtual cactus groups, respectively. The degeneration of a twisted real form of the Deligne-Mumford space to $\overline F_n(\mathbb{R})$ gives rise to a natural homomorphism from the affine cactus group to the virtual cactus group.

math.AG↗
arXiv2023

A Pair of Garside Shadows

We prove that the smallest elements of Shi parts and cone type parts exist and form Garside shadows. The latter resolves a conjecture of Parkinson and the second author as well as a conjecture of Hohlweg, Nadeau and Williams.

math.GR↗
arXiv2022

Coxeter groups are biautomatic

We prove that Coxeter groups are biautomatic. From our construction of the biautomatic structure it follows that uniform lattices in isometry groups of buildings are biautomatic.

math.GR↗
arXiv2022

Torsion groups do not act on $2$-dimensional $\mathrm{CAT}(0)$ complexes

We show, under mild hypotheses, that if each element of a finitely generated group acting on a $2$-dimensional $\mathrm{CAT}(0)$ complex has a fixed point, then there is a global fixed point. In particular all actions of finitely generated torsion groups on such complexes have global fixed points. The proofs rely on Masur's theorem on periodic trajectories in rational billiards, and Ballmann-Brin's methods for finding closed geodesics in $2$-dimensional locally $\mathrm{CAT}(0)$ complexes. As another ingredient we prove that the image of an immersed loop in a graph of girth $2π$ with length not commensurable with $π$ has diameter $> π$. This is closely related to a theorem of Dehn on tiling rectangles by squares.

math.GR↗
arXiv2021

Tits Alternative for groups acting properly on $2$-dimensional recurrent complexes (with an appendix written jointly with Jon McCammond)

We prove the Tits Alternative for groups acting on $2$-dimensional "recurrent" complexes with uniformly bounded cell stabilisers. This class of complexes includes, among others: $2$-dimensional Euclidean buildings, $2$-dimensional systolic complexes, $B(6)$-small cancellation complexes, and standard Cayley complexes for Artin groups of extra-large type. In the appendix written jointly with Jon McCammond we extend the result to a class of $2$-dimensional Artin groups containing all large-type Artin groups.

math.GR↗
arXiv2021

Unicorn paths and hyperfiniteness for the mapping class group

Let S be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of orbit equivalence relations induced by the actions of the mapping class group of S on the Gromov boundaries of the arc graph and the curve graph of S. In the curve graph case, this strengthens the results of Hamenstädt and Kida that this action is universally amenable and that the mapping class group of S is exact.

math.GT↗
arXiv2021

Dihedral twists in the Twist Conjecture

Under the assumption that a defining graph of a Coxeter group admits only subsequent elementary twists in $\mathbb{Z}_2$ or dihedral groups and is of type $\mathrm{FC}$, we prove Bernhard Mühlherr's Twist Conjecture

math.GR↗
arXiv2021

Acylindrical actions for two-dimensional Artin groups of hyperbolic type

For a two-dimensional Artin group $A$ whose associated Coxeter group is hyperbolic, we prove that the action of $A$ on the hyperbolic space obtained by coning off certain subcomplexes of its modified Deligne complex is acylindrical. Moreover, if for each $s\in S$ there is $t\in S$ with $m_{st}< \infty$, then this action is universal. As a consequence, for $|S|\geq 3$, if $A$ is irreducible, then it is acylindrically hyperbolic. We also obtain the Tits alternative for $A$, and we classify the subgroups of $A$ that virtually split as a direct product. A key ingredient in our approach is a simple criterion to show the acylindricity of an action on a two-dimensional $\mathrm{CAT}(-1)$ complex.

math.GR↗
arXiv2021

2-dimensional Coxeter groups are biautomatic

Let $W$ be a $2$-dimensional Coxeter group, that is, a one with $\frac{1}{m_{st}}+\frac{1}{m_{sr}}+\frac{1}{m_{tr}}\leq 1$ for all triples of distinct $s,t,r\in S$. We prove that $W$ is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary $W$), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type $W$ are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\widetilde{A}_3$.

math.GR↗
arXiv2019

Presqu'un immeuble pour le groupe des automorphismes modérés

Inspired by the Bruhat-Tits building of SL$_n$($\mathbb Q_p$), we construct a complete metric space X with an action of the tame automorphism group of the affine space Tame($K^n$). The points in X are certain monomial valuations, and X admits a natural structure of Euclidean CW-complex of dimension n-1. When n = 3, and for K of characteristic zero, we prove that X has non-positive curvature and is simply connected, hence is a CAT(0) space. As an application we obtain the linearizability of finite subgroups in Tame($K^3$).

math.GR↗
arXiv2019

Tits alternative for Artin groups of type FC

Given a group action on a finite-dimensional CAT(0) cube complex, we give a simple criterion phrased purely in terms of cube stabilisers that ensures that the group satisfies the strong Tits alternative, provided that each vertex stabiliser satisfies the strong Tits alternative. We use it to prove that all Artin groups of type FC satisfy the strong Tits alternative.

math.GR↗