Group Theory
- [1] arXiv:2405.09722 [pdf, ps, html, other]
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Title: Embedding finitely presented self-similar groups into finitely presented simple groupsComments: 10 pagesSubjects: Group Theory (math.GR)
We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone-Higman conjecture. The simple groups in question are certain commutator subgroups of Röver-Nekrashevych groups, and the difficulty lies in the fact that even if a Röver-Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Q})$ satisfies the Boone-Higman conjecture.
- [2] arXiv:2405.09736 [pdf, ps, html, other]
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Title: On the conjugacy separability of ordinary and generalized Baumslag-Solitar groupsComments: 14 pages, in Russian; an English version will be available soonSubjects: Group Theory (math.GR)
Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism $\sigma$ of $X$ onto a group from $\mathcal{C}$ such that the elements $x\sigma$ and $y\sigma$ are still not equal (respectively, not conjugate in $X\sigma$). A generalized Baumslag-Solitar group or GBS-group is the fundamental group of a finite connected graph of groups whose all vertex and edge groups are infinite cyclic. An ordinary Baumslag-Solitar group is the GBS-group that corresponds to a graph containing only one vertex and one loop. Suppose that the class $\mathcal{C}$ consists of periodic groups and is closed under taking subgroups and unrestricted wreath products. We prove that a non-solvable GBS-group is conjugacy $\mathcal{C}$-separable if and only if it is residually a $\mathcal{C}$-group. We also find a criterion for a solvable GBS-group to be conjugacy $\mathcal{C}$-separable. As a corollary, we prove that an arbitrary GBS-group is conjugacy (finite) separable if and only if it is residually finite.
- [3] arXiv:2405.10136 [pdf, ps, html, other]
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Title: The automorphism tower of the Mennicke group $M(-1,-1,-1)$Subjects: Group Theory (math.GR)
We compute the automorphism tower of the centerless Mennicke group $M(-1,-1,-1)$
- [4] arXiv:2405.10147 [pdf, ps, html, other]
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Title: Isomorphism of relative holomorphs and matrix similaritySubjects: Group Theory (math.GR)
Let $V$ be a finite-dimensional vector space over the field with $p$ elements, where $p$ is a prime number. Given arbitrary $\alpha,\beta\in \mathrm{GL}(V)$, we consider the simidirect products $V\rtimes\langle \alpha\rangle$ and $V\rtimes\langle \beta\rangle$, and show that if $V\rtimes\langle \alpha\rangle$ and $V\rtimes\langle \beta\rangle$ are isomorphic, then $\alpha$ must be similar to a power of $\beta$ that generates the same subgroup as $\beta$; that is, if $H$ and $K$ are cyclic subgroups of $\mathrm{GL}(V)$ such that $V\rtimes H\cong V\rtimes K$, then $H$ and $K$ must be conjugate subgroups of $\mathrm{GL}(V)$. If we remove the cyclic condition, there exist examples of non-isomorphic, let alone non-conjugate, subgroups $H$ and $K$ of $\mathrm{GL}(V)$ such that $V\rtimes H\cong V\rtimes K$. Even if we require that non-cyclic subgroups $H$ and $K$ of $\mathrm{GL}(V)$ be abelian, we may still have $V\rtimes H\cong V\rtimes K$ with $H$ and $K$ non-conjugate in $\mathrm{GL}(V)$, but in this case, $H$ and $K$ must at least be isomorphic. If we replace $V$ by a free module $U$ over ${\mathbf Z}/p^m{\mathbf Z}$ of finite rank, with $m>1$, it may happen that $U\rtimes H\cong U\rtimes K$ for non-conjugate cyclic subgroups of $\mathrm{GL}(U)$. If we completely abandon our requirements on $V$, a sufficient criterion is given for a finite group $G$ to admit non-conjugate cyclic subgroups $H$ and $K$ of $\mathrm{Aut}(G)$ such that $G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.
- [5] arXiv:2405.10172 [pdf, ps, html, other]
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Title: Parallel Hopf-Galois structures on separable field extensionsComments: 17 pagesSubjects: Group Theory (math.GR)
Let $L/K$ be a finite separable extension of fields of degree $n$, and let $E/K$ be its Galois closure. Greither and Pareigis showed how to find all Hopf-Galois structures on $L/K$. We will call a subextension $L'/K$ of $E/K$ \textit{parallel} to $L/K$ if $[L':K]=n$.
In this paper, we investigate the relationship between the Hopf-Galois structures on an extension $L/K$ and those on its parallel extensions. We give an example of a transitive subgroup corresponding to an extension admitting a Hopf-Galois structure but that has a parallel extension admitting no Hopf-Galois structures. We show that once one has such a situation, it can be extended into an infinite family of transitive subgroups admitting this phenomenon. We also investigate this fully in the case of extensions of degree $pq$ with $p,q$ distinct odd primes, and show that there is no example of such an extension admitting the phenomenon. - [6] arXiv:2405.10234 [pdf, ps, html, other]
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Title: Boone--Higman Embeddings for Contracting Self-Similar GroupsComments: 6 page, no figuresSubjects: Group Theory (math.GR)
We give a short proof that every contracting self-similar group embeds into a finitely presented simple group. In particular, any contracting self-similar group embeds into the corresponding Röver--Nekrashevych group, and this in turn embeds into one of the twisted Brin--Thompson groups introduced by the first author and Matthew Zaremsky. The proof here is a simplification of a more general argument given by the authors, Collin Bleak, and Matthew Zaremsky for contracting rational similarity groups.
New submissions for Friday, 17 May 2024 (showing 6 of 6 entries )
- [7] arXiv:2405.09710 (cross-list from math.CO) [pdf, ps, html, other]
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Title: The lattice of submonoids of the uniform block permutations containing the symmetric groupComments: 15 pagesSubjects: Combinatorics (math.CO); Group Theory (math.GR)
We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.
- [8] arXiv:2405.09748 (cross-list from q-bio.CB) [pdf, ps, html, other]
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Title: A Mathematical Reconstruction of Endothelial Cell NetworksComments: 14 pages, 9 figuresSubjects: Cell Behavior (q-bio.CB); Combinatorics (math.CO); Group Theory (math.GR); Quantitative Methods (q-bio.QM)
Endothelial cells form the linchpin of vascular and lymphatic systems, creating intricate networks that are pivotal for angiogenesis, controlling vessel permeability, and maintaining tissue homeostasis. Despite their critical roles, there is no rigorous mathematical framework to represent the connectivity structure of endothelial networks. Here, we develop a pioneering mathematical formalism called $\pi$-graphs to model the multi-type junction connectivity of endothelial networks. We define $\pi$-graphs as abstract objects consisting of endothelial cells and their junction sets, and introduce the key notion of $\pi$-isomorphism that captures when two $\pi$-graphs have the same connectivity structure. We prove several propositions relating the $\pi$-graph representation to traditional graph-theoretic representations, showing that $\pi$-isomorphism implies isomorphism of the corresponding unnested endothelial graphs, but not vice versa. We also introduce a temporal dimension to the $\pi$-graph formalism and explore the evolution of topological invariants in spatial embeddings of $\pi$-graphs. Finally, we outline a topological framework to represent the spatial embedding of $\pi$-graphs into geometric spaces. The $\pi$-graph formalism provides a novel tool for quantitative analysis of endothelial network connectivity and its relation to function, with the potential to yield new insights into vascular physiology and pathophysiology.
- [9] arXiv:2405.10021 (cross-list from math.RT) [pdf, ps, html, other]
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Title: $\tau$-Tilting finiteness of group algebras of semidirect products of abelian $p$-groups and abelian $p'$-groupsComments: 16 pagesSubjects: Representation Theory (math.RT); Group Theory (math.GR); Rings and Algebras (math.RA)
Demonet, Iyama and Jasso introduced a new class of finite dimensional algebras, $\tau$-tilting finite algebras. It was shown by Eisele, Janssens and Raedschelders that tame blocks of group algebras of finite groups are always $\tau$-tilting finite. Given the classical result that the representation type (representation finite, tame or wild) of blocks is determined by their defect groups, it is natural to ask what kinds of subgroups control $\tau$-tilting finiteness of group algebras or their blocks. In this paper, as a positive answer to this question, we demonstrate that $\tau$-tilting finiteness of a group algebra of a finite group $G$ is controlled by a $p$-hyperfocal subgroup of $G$ under some assumptions on $G$. We consider a group algebra of a finite group $P\rtimes H$ over an algebraically closed field of positive characteristic $p$, where $P$ is an abelian $p$-group and $H$ is an abelian $p'$-group acting on $P$, and show that $p$-hyperfocal subgroups determine $\tau$-tilting finiteness of the group algebras in this case.
- [10] arXiv:2405.10159 (cross-list from math.NT) [pdf, ps, html, other]
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Title: Artin-Schreier towers of finite fieldsSubjects: Number Theory (math.NT); Group Theory (math.GR)
Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where $c_i$ is a root of $X^p-X-a_{i-1}$ and $a_{i-1}=(c_{-1}\cdots c_{i-1})^{p-1}$, with $c_{-1}=1$. We extend and strengthen to arbitrary primes prior work of Popovych for $p=2$ on the multiplicative order of the given generator $c_i$ for $L_i$ over $L_{i-1}$. In particular, for $i\geq 0$, we show that $O(c_i)=O(a_i)$, except only when $p=2$ and $i=1$, and that $O(c_i)$ is equal to the product of the orders of $c_j$ modulo $L_{j-1}^\times$, where $0\leq j\leq i$ if $p$ is odd, and $i\geq 2$ and $1\leq j\leq i$ if $p=2$. We also show that for $i\geq 0$, the $\mathrm{Gal}(L_i/L_{i-1})$-conjugates of $a_i$ form a normal basis of $L_i$ over $L_{i-1}$. In addition, we obtain the minimal polynomial of $c_1$ over $F_p$ in explicit form.
- [11] arXiv:2405.10191 (cross-list from math.OA) [pdf, ps, html, other]
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Title: Amenable actions on ill-behaved simple C*-algebrasComments: 17 pagesSubjects: Operator Algebras (math.OA); Group Theory (math.GR)
By combining Rørdam's construction and author's previous construction, we provide the first examples of amenable actions on simple separable nuclear C*-algebras that are neither stable finite nor purely infinite. For free groups, we also provide unital examples. We arrange the actions so that the crossed products are still simple with both a finite and an infinite projection.
- [12] arXiv:2405.10209 (cross-list from math.GT) [pdf, ps, html, other]
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Title: Remarks on discrete subgroups with full limit sets in higher rank Lie groupsComments: Comments are welcome!Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS); Group Theory (math.GR)
We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of $G = \operatorname{SL}(3,\mathbb{R})$ must have a full limit set in the Furstenberg boundary of $G$.
In the appendix, we show the the existence of Zariski-dense discrete subgroups $\Gamma$ of $\operatorname{SL}(n,\mathbb{R})$, where $n\ge 3$, such that the Jordan projection of some loxodromic element $\gamma \in\Gamma$ lies on the boundary of the limit cone of $\Gamma$.
Cross submissions for Friday, 17 May 2024 (showing 6 of 6 entries )
- [13] arXiv:2311.06853 (replaced) [pdf, ps, html, other]
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Title: A note on the existence of the Reidemeister zeta function on groupsComments: 15 pages - new version incorporates referee commentsSubjects: Group Theory (math.GR)
Given an endomorphism $\varphi: G \to G$ on a group $G$, one can define the Reidemeister number $R(\varphi) \in \mathbb{N} \cup \{\infty\}$ as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function $R_\varphi(z)$, by using the Reidemeister numbers $R(\varphi^n)$ of iterates $\varphi^n$ in order to define a power series, has been studied a lot in the literature, especially the question whether it is a rational function or not. For example, it has been shown that the answer is positive for finitely generated torsion-free virtually nilpotent groups, but negative in general for abelian groups that are not finitely generated.
However, in order to define the Reidemeister zeta function of an endomorphism $\varphi$, it is necessary that the Reidemeister numbers $R(\varphi^n)$ of all iterates $\varphi^n$ are finite. This puts restrictions, not only on the endomorphism $\varphi$, but also on the possible groups $G$ if $\varphi$ is assumed to be injective. In this note, we want to initiate the study of groups having a well-defined Reidemeister zeta function for a monomorphism $\varphi$, because of its importance for describing the behavior of Reidemeister zeta functions. As a motivational example, we show that the Reidemeister zeta function is indeed rational on torsion-free virtually polycyclic groups. Finally, we give some partial results about the existence in the special case of automorphisms on finitely generated torsion-free nilpotent groups, showing that it is a restrictive condition. - [14] arXiv:2202.00052 (replaced) [pdf, ps, html, other]
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Title: On flat manifold bundles and the connectivity of Haefliger's classifying spacesComments: This version is to appear in the Proceedings of the AMSSubjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)
We investigate a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In this context, Haefliger-Thurston's conjecture predicts that every $M$-bundle over a manifold $B$ where $\text{dim}(B)\leq \text{dim}(M)$ is cobordant to a flat $M$-bundle. In particular, we study the bordism class of flat $M$-bundles over low dimensional manifolds, comparing a finite dimensional Lie group $G$ with $\text{Diff}_0(G)$ and localizing the holonomy of flat M-bundles to be supported in a ball.
- [15] arXiv:2305.16880 (replaced) [pdf, ps, other]
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Title: On the first order theory of plactic monoidsComments: Main changes to version 2: new section added with new bi-interpretability results. Previous results in sections 3, 4, and 6 (previously section 5) unchanged, but have been rewritten in a clearer style. Abstract rewritten to reflect new results. Errors and typos in bibliography fixedSubjects: Logic (math.LO); Group Theory (math.GR)
This paper proves that a plactic monoid of any finite rank will have decidable first order theory. This resolves other open decidability problems about the finite rank plactic monoids, such as the Diophantine problem and identity checking. This is achieved by interpreting a plactic monoid of arbitrary rank in Presburger arithmetic, which is known to have decidable first order theory. We also prove that the interpretation of the plactic monoids into Presburger Arithmetic is in fact a bi-interpretation, hence any two plactic monoids of finite rank are bi-interpretable with one another. The algorithm generating the interpretations is uniform, which answers positively the decidability of the Diophantine problem for the infinite rank plactic monoid.
- [16] arXiv:2312.05663 (replaced) [pdf, ps, html, other]
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Title: Bridging colorings of virtual links from virtual biquandles to biquandlesComments: 12 pages, title changed, references updatedSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
A biquandle is a solution to the set-theoretical Yang-Baxter equation, which yields invariants for virtual knots such as the coloring number and the state-sum invariant. A virtual biquandle enriches the structure of a biquandle by incorporating an invertible unary map. This unary operator plays a crucial role in defining the action of virtual crossings on the labels of incoming arcs in a virtual link diagram. This leads to extensions of invariants from biquandles to virtual biquandles, thereby enhancing their strength.
In this article, we establish a connection between the coloring invariant derived from biquandles and virtual biquandles. We prove that the number of colorings of a virtual link $L$ by virtual biquandles can be recovered from colorings by biquandles. We achieve this by proving the equivalence between two different representations of virtual braid groups. Furthermore, we introduce a new set of labeling rules using which one can construct a presentation of the associated fundamental virtual biquandle of $L$ using only the relations coming from the classical crossings. This is an improvement to the traditional method, where writing down a presentation of the associated fundamental virtual biquandle necessitates noting down the relations arising from the classical and virtual crossings. - [17] arXiv:2402.02312 (replaced) [pdf, ps, html, other]
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Title: Higher Congruences in Character TablesComments: 12 pages. Comments welcome!Subjects: Representation Theory (math.RT); Combinatorics (math.CO); Group Theory (math.GR)
Motivated by recent work of Peluse and Soundararajan on divisibility properties of the entries of the character tables of symmetric groups, we investigate the question: For a finite group G, when are two columns of the character table of G congruent to one another modulo a power of a prime?