arXiv subjects

Stavroula Makri

Publications and source records attributed to Stavroula Makri.

6 recordsLinked to original sources

arXiv2026

The generalized Lefschetz number and loop braid groups

We study the interplay between braid group theory and topological dynamics in three dimensions. While classical braid theory has been extensively applied to surface homeomorphisms to analyze fixed and periodic points, an analogous framework in three-dimensional manifolds has been lacking. In this work, we introduce the use of loop braid groups as a three-dimensional generalization of classical braid groups to investigate homeomorphisms of the 3-ball that leave invariant a finite collection of circles. In our main theorem we associate the Burau matrix representations of loop braid elements to the generalized Lefschetz number. This result provides important information on the existence and interaction of fixed and periodic points of such homeomorphisms. In addition, an application of our theorem gives an estimate of the number of their periodic points. Our theorem establishes a three-dimensional analogue of a classical result, providing the first framework that connects loop braid groups with Nielsen fixed point theory and topological dynamics in dimension three, providing a rich 3-dimensional framework, whose topological and algebraic aspects have been extensively investigated, for studying its topological dynamical properties.

math.GT↗

arXiv2025

On sections of configurations of points on orientable surfaces

We study the configuration space of distinct, unordered points on compact orientable surfaces of genus $g$, denoted $S_g$. Specifically, we address the section problem, which concerns the addition of $n$ distinct points to an existing configuration of $m$ distinct points on $S_g$ in a way that ensures the new points vary continuously with respect to the initial configuration. This problem is equivalent to the splitting problem in surface braid groups. With an algebraic approach, for $g\geq 1$ and $m\geq 2$, we establish a necessary condition for the existence of a section, showing that if a section exists, then $n$ must be a multiple of $m+(2g-2)$. For $g\geq 1$ and $m=1$, we take a geometric approach to demonstrate that a section exists for all values of $n$.

math.GT↗

arXiv2024

Strong Nielsen equivalence on the punctured disc

Let $f$ be an orientation-preserving homeomorphism of the 2-disc $\mathbb{D}^2$ that fixes the boundary pointwise and leaves invariant a finite subset in the interior of $\mathbb{D}^2$. We study the strong Nielsen equivalence of periodic points of such homeomorphisms $f$ and we give a necessary and sufficient condition for two periodic points to be strong Nielsen equivalent in the context of braid theory. In addition, we present an application of our result to the trace formula given by Jiang--Zheng, deducing that the obtained forced periodic orbits belong to different strong Nielsen classes.

math.DS↗

arXiv2022

The unrestricted virtual braid groups $UVB_n$

Let $UVB_n$ and $UVP_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups $UVB_n$ and $UVP_n$, and our main results are as follows: for $n\geq 5$, we give a complete description, up to conjugation, to all possible homomorphisms from $UVB_n$ to the symmetric group $S_n$. For $n\geq 3$, we characterise all possible images of $UVB_n$, under a group homomorphism, to any finite group $G$. For $n\geq 5$, we prove that $UVP_n$ is a characteristic subgroup of $UVB_n$. In addition, we determine the automorphism group of $UVP_n$ and we prove that $\mathbb{Z}_2\times\mathbb{Z}_2$ is a subgroup of the outer automorphism group of $UVB_n$. Lastly, we show that $UVB_n$ and $UVP_n$ are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group $WB_n$ and we discuss about its automorphism group.

math.GT↗

arXiv2022

The braid groups $B_{n,m}(\mathbb{R}P^2)$ and the splitting problem of the generalised Fadell-Neuwirth short exact sequence

Let $n,m\in \mathbb{N}$, and let $B_{n,m}(\mathbb{R}P^2)$ be the set of $(n + m)$-braids of the projective plane whose associated permutation lies in the subgroup $S_n\times S_m$ of the symmetric group $S_{n+m}$. We study the splitting problem of the following generalisation of the Fadell-Neuwirth short exact sequence: $$1\rightarrow B_m(\mathbb{R}P^2 \setminus \{x_1,\dots,x_n\})\rightarrow B_{n,m}(\mathbb{R}P^2)\xrightarrow{\bar{q}} B_n(\mathbb{R}P^2)\rightarrow 1,$$ where the map $\bar{q}$ can be considered geometrically as the epimorphism that forgets the last $m$ strands, as well as the existence of a section of the corresponding fibration $q:F_{n+m}(\mathbb{R}P^2)/S_n\times S_m\to F_{n}(\mathbb{R}P^2)/S_n$, where we denote by $F_n(\mathbb{R}P^2)$ the $n^{th}$ ordered configuration space of the projective plane $\mathbb{R}P^2$. Our main results are the following: if $n=1$ the homomorphism $\bar{q}$ and the corresponding fibration $q$ admits no section, while if $n=2$, then $\bar{q}$ and $q$ admit a section. For $n\geq 3$, we show that if $\bar{q}$ and $q$ admit a section then $m\equiv 0, (n-1)^2\ \textrm{mod}\ n(n-1)$. Moreover, using geometric constructions, we show that the homomorphism $\bar{q}$ and the fibration $q$ admit a section for $m=kn(2n-1)(2n-2)$, where $ k\geq1$, and for $m=2n(n-1)$. In addition, we show that for $m\geq3$, $B_m(\mathbb{R}P^2\setminus\{x_1,\dots,x_n\})$ is not residually nilpotent and for $m\geq 5$, it is not residually solvable.

math.GT↗

arXiv2022

Unrestricted virtual braids and crystallographic braid groups

We show that the crystallographic braid group $B_n/[P_n,P_n]$ embeds naturally in the group of unrestricted virtual braids $UVB_n$, we give new proofs of known results about the torsion elements of $B_n/[P_n,P_n]$, and we characterise the torsion elements of $UVB_n$.

math.GR↗