$m$-location is a local combinatorial condition for flag simplicial complexes introduced by Osajda. Osajda showed that simply connected 8-located locally 5-large complexes are hyperbolic. We treat the nonpositive curvature case of 7-located locally 5-large complexes. We show that any minimal area disc diagram in a 7-located locally 5-large complex is itself 7-located and locally 5-large. We define a natural CAT(0) metric for 7-located disc diagrams and use this to prove that simply connected 7-located locally 5-large complexes have quadratic isoperimetric function. Along the way, we prove that locally weakly systolic complexes are 7-located locally 5-large.
{"title":"7-location, weak systolicity and isoperimetry","authors":"Nima Hoda, Ioana-Claudia Lazăr","doi":"arxiv-2409.00612","DOIUrl":"https://doi.org/arxiv-2409.00612","url":null,"abstract":"$m$-location is a local combinatorial condition for flag simplicial complexes\u0000introduced by Osajda. Osajda showed that simply connected 8-located locally\u00005-large complexes are hyperbolic. We treat the nonpositive curvature case of\u00007-located locally 5-large complexes. We show that any minimal area disc diagram in a 7-located locally 5-large\u0000complex is itself 7-located and locally 5-large. We define a natural CAT(0)\u0000metric for 7-located disc diagrams and use this to prove that simply connected\u00007-located locally 5-large complexes have quadratic isoperimetric function.\u0000Along the way, we prove that locally weakly systolic complexes are 7-located\u0000locally 5-large.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered string of terms from $G$ with repetition allowed, and we say that it is a product-one sequence if its terms can be ordered so that their product is the identity element of $G$. Then, the Davenport constant $mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is a product-one sequence which cannot be partitioned into two non-trivial product-one subsequences. The Davenport constant is a combinatorial group invariant that has been studied fruitfully over several decades in additive combinatorics, invariant theory, and factorization theory, etc. Apart from a few cases of finite groups, the precise value of the Davenport constant is unknown. Even in the abelian case, little is known beyond groups of rank at most two. On the other hand, for a fixed positive integer $r$, structural results characterizing which groups $G$ satisfy $mathsf D (G) = r$ are rare. We only know that there are finitely many such groups. In this paper, we study the classification of finite groups based on the Davenport constant.
让 $G$ 是一个有限群。我们所说的$G$上的序列是指$G$中允许重复的有限无序项串，如果它的项可以有序排列，使得它们的乘积是$G$的同元素，我们就说它是乘积一序列。那么，达文波特常数 $mathsf D (G)$ 是最小积一序列的最大长度，即一个积一序列不能被分割成两个非三积一子序列。达文波特常数是一个组合群不变式，几十年来在加法组合学、不变式理论和因式分解理论等方面进行了卓有成效的研究。除了无穷群的少数情况外，达文波特常数的精确值尚属未知。即使是无边群，除了秩最多为 2 的群之外，其他群也鲜为人知。另一方面，对于固定的正整数 $r$，描述哪些群 $G$ 满足 $mathsf D (G) = r$ 的结构性结果也很罕见。我们只知道有有限多个这样的群。本文研究了基于达文波特常数的无限群分类。
{"title":"A classification of finite groups with small Davenport constant","authors":"Jun Seok Oh","doi":"arxiv-2409.00363","DOIUrl":"https://doi.org/arxiv-2409.00363","url":null,"abstract":"Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered\u0000string of terms from $G$ with repetition allowed, and we say that it is a\u0000product-one sequence if its terms can be ordered so that their product is the\u0000identity element of $G$. Then, the Davenport constant $mathsf D (G)$ is the\u0000maximal length of a minimal product-one sequence, that is a product-one\u0000sequence which cannot be partitioned into two non-trivial product-one\u0000subsequences. The Davenport constant is a combinatorial group invariant that\u0000has been studied fruitfully over several decades in additive combinatorics,\u0000invariant theory, and factorization theory, etc. Apart from a few cases of\u0000finite groups, the precise value of the Davenport constant is unknown. Even in\u0000the abelian case, little is known beyond groups of rank at most two. On the\u0000other hand, for a fixed positive integer $r$, structural results characterizing\u0000which groups $G$ satisfy $mathsf D (G) = r$ are rare. We only know that there\u0000are finitely many such groups. In this paper, we study the classification of\u0000finite groups based on the Davenport constant.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider two families of cyclic presentations and show that, subject to certain conditions on the defining parameters, they are spines of closed 3-manifolds. For the first family, the Whitehead graphs have not previously been observed in this context, and the corresponding manifolds are lens spaces. The second family provides new examples where the reduced Whitehead graphs are those of the Fractional Fibonacci presentations; here the corresponding manifolds are often (but not always) hyperbolic.
{"title":"3-manifold spine cyclic presentations with seldom seen Whitehead graphs","authors":"Gerald Williams","doi":"arxiv-2408.17125","DOIUrl":"https://doi.org/arxiv-2408.17125","url":null,"abstract":"We consider two families of cyclic presentations and show that, subject to\u0000certain conditions on the defining parameters, they are spines of closed\u00003-manifolds. For the first family, the Whitehead graphs have not previously\u0000been observed in this context, and the corresponding manifolds are lens spaces.\u0000The second family provides new examples where the reduced Whitehead graphs are\u0000those of the Fractional Fibonacci presentations; here the corresponding\u0000manifolds are often (but not always) hyperbolic.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206479","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study several combinatorial properties of finite groups that are related to the notions of sequenceability, R-sequenceability, and harmonious sequences. In particular, we show that in every abelian group $G$ with a unique involution $imath_G$ there exists a permutation $g_0,ldots, g_{m}$ of elements of $G backslash {imath_G}$ such that the consecutive sums $g_0+g_1, g_1+g_2,ldots, g_{m}+g_0$ also form a permutation of elements of $Gbackslash {imath_G}$. We also show that in every abelian group of order at least 4 there exists a sequence containing each non-identity element of $G$ exactly twice such that the consecutive sums also contain each non-identity element of $G$ twice. We apply several results to the existence of transversals in Latin squares.
{"title":"Harmonious sequences in groups with a unique involution","authors":"Mohammad Javaheri, Lydia de Wolf","doi":"arxiv-2408.16207","DOIUrl":"https://doi.org/arxiv-2408.16207","url":null,"abstract":"We study several combinatorial properties of finite groups that are related\u0000to the notions of sequenceability, R-sequenceability, and harmonious sequences.\u0000In particular, we show that in every abelian group $G$ with a unique involution\u0000$imath_G$ there exists a permutation $g_0,ldots, g_{m}$ of elements of $G\u0000backslash {imath_G}$ such that the consecutive sums $g_0+g_1,\u0000g_1+g_2,ldots, g_{m}+g_0$ also form a permutation of elements of $Gbackslash\u0000{imath_G}$. We also show that in every abelian group of order at least 4\u0000there exists a sequence containing each non-identity element of $G$ exactly\u0000twice such that the consecutive sums also contain each non-identity element of\u0000$G$ twice. We apply several results to the existence of transversals in Latin\u0000squares.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article investigates neighborhoods' sizes in the enhanced power graph (as known as the cyclic graph) associated with a finite group. In particular, we characterize finite $p$-groups with the smallest maximum size for neighborhoods of nontrivial element in its enhanced power graph.
{"title":"On neighborhoods in the enhanced power graph associated with a finite group","authors":"Mark L. Lewis, Carmine Monetta","doi":"arxiv-2408.16545","DOIUrl":"https://doi.org/arxiv-2408.16545","url":null,"abstract":"This article investigates neighborhoods' sizes in the enhanced power graph\u0000(as known as the cyclic graph) associated with a finite group. In particular,\u0000we characterize finite $p$-groups with the smallest maximum size for\u0000neighborhoods of nontrivial element in its enhanced power graph.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we define a new hypergraph $mathcal{H(V,E)}$ on a loop $L$, where $mathcal{V}$ is the set of points of the loop $L$ and $mathcal{E}$ is the set of hyperedges $e={x,y,z}$ such that $x,y$ and $z$ associate in the order they are written. We call this hypergraph as the associating hypergraph on a loop $L$. We study certain properites of associating hypergraphs on the Moufang loop $M(D_n,2)$, where $D_n$ denotes the dihedral group of order $2n$.
{"title":"Associating hypergraphs defined on loops","authors":"Siddharth Malviy, Vipul Kakkar","doi":"arxiv-2408.16459","DOIUrl":"https://doi.org/arxiv-2408.16459","url":null,"abstract":"In this paper, we define a new hypergraph $mathcal{H(V,E)}$ on a loop $L$,\u0000where $mathcal{V}$ is the set of points of the loop $L$ and $mathcal{E}$ is\u0000the set of hyperedges $e={x,y,z}$ such that $x,y$ and $z$ associate in the\u0000order they are written. We call this hypergraph as the associating hypergraph\u0000on a loop $L$. We study certain properites of associating hypergraphs on the\u0000Moufang loop $M(D_n,2)$, where $D_n$ denotes the dihedral group of order $2n$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we determine the non-real elements--the ones that are not conjugate to their inverses--in the group $G = G_2(q)$ when $char(F_q)neq 2,3$. We use this to show that this group is chiral; that is, there is a word w such that $w(G)neq w(G)^{-1}$. We also show that most classical finite simple groups are achiral
{"title":"Chirality and non-real elements in $G_2(q)$","authors":"Sushil Bhunia, Amit Kulshrestha, Anupam Singh","doi":"arxiv-2408.15546","DOIUrl":"https://doi.org/arxiv-2408.15546","url":null,"abstract":"In this article, we determine the non-real elements--the ones that are not\u0000conjugate to their inverses--in the group $G = G_2(q)$ when $char(F_q)neq\u00002,3$. We use this to show that this group is chiral; that is, there is a word w\u0000such that $w(G)neq w(G)^{-1}$. We also show that most classical finite simple\u0000groups are achiral","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ilaria Castellano, Bianca Marchionna, Thomas Weigel
In several instances, the invariants of compactly generated totally disconnected locally compact groups acting on locally finite buildings can be conveniently described via invariants of the Coxeter group representing the type of the building. For certain totally disconnected locally compact groups acting on buildings, we establish and collect several results concerning, for example, the rational discrete cohomological dimension (cf. Thm. A), the flat-rank (cf. Thm. C) and the number of ends (cf. Cor. K). Moreover, for an arbitrary compactly generated totally disconnected locally compact group, we express the number of ends in terms of its cohomology groups (cf. Thm. J). Furthermore, generalising a result of F. Haglund and F. Paulin, we prove that visual graph of groups decompositions of a Coxeter group $(W,S)$ can be used to construct trees from buildings of type $(W,S)$. We exploit the latter result to show that all $sigma$-compact totally disconnected locally compact groups acting chamber-transitively on buildings can be decomposed accordingly to any visual graph of groups decomposition of the type $(W,S)$ (cf. Thm. F and Cor. G).
在某些情况下，作用于局部有限建筑物的紧凑生成的完全互不相连局部紧凑群的不变式可以通过代表建筑物类型的考斯特群的不变式来方便地描述。对于作用于建筑物的某些完全互不相连的局部紧凑群，我们建立并收集了几个结果，例如，合理离散同调维数（参见 Thm.A）、平秩（参见 Thm.C）和端数（参见 Cor.K）。此外，对于一个任意紧凑生成的完全断开局部紧凑群，我们用它的同调群来表达末端数（参见 Thm.J ）。此外，我们推广了 F. Haglund 和 F. Paulin 的一个结果，证明考斯特群 $(W,S)$ 的可视群图分解可以用来从类型 $(W,S)$ 的建筑物中构造树。我们利用后一个结果来证明，所有对建筑物起室反作用的$sigma$-compact totally disconnected locally compact群，都可以相应地分解为类型为$(W,S)$的任何可见群分解图（参见定理F和定理G）。
{"title":"Weyl-invariants of totally disconnected locally compact groups acting cocompactly on buildings","authors":"Ilaria Castellano, Bianca Marchionna, Thomas Weigel","doi":"arxiv-2408.15716","DOIUrl":"https://doi.org/arxiv-2408.15716","url":null,"abstract":"In several instances, the invariants of compactly generated totally\u0000disconnected locally compact groups acting on locally finite buildings can be\u0000conveniently described via invariants of the Coxeter group representing the\u0000type of the building. For certain totally disconnected locally compact groups\u0000acting on buildings, we establish and collect several results concerning, for\u0000example, the rational discrete cohomological dimension (cf. Thm. A), the\u0000flat-rank (cf. Thm. C) and the number of ends (cf. Cor. K). Moreover, for an\u0000arbitrary compactly generated totally disconnected locally compact group, we\u0000express the number of ends in terms of its cohomology groups (cf. Thm. J).\u0000Furthermore, generalising a result of F. Haglund and F. Paulin, we prove that\u0000visual graph of groups decompositions of a Coxeter group $(W,S)$ can be used to\u0000construct trees from buildings of type $(W,S)$. We exploit the latter result to\u0000show that all $sigma$-compact totally disconnected locally compact groups\u0000acting chamber-transitively on buildings can be decomposed accordingly to any\u0000visual graph of groups decomposition of the type $(W,S)$ (cf. Thm. F and Cor.\u0000G).","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly $mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple Lie groups of higher rank are not strictly $mathbb{C}$-stable. Furthermore, we prove that free groups are not uniformly flexibly $F$-stable over any field $F$.
{"title":"Uniform rank metric stability of Lie algebras, Lie groups and lattices","authors":"Benjamin Bachner","doi":"arxiv-2408.15614","DOIUrl":"https://doi.org/arxiv-2408.15614","url":null,"abstract":"We study uniform stability of discrete groups, Lie groups and Lie algebras in\u0000the rank metric, and the connections between uniform stability of these\u0000objects. We prove that semisimple Lie algebras are far from being flexibly\u0000$mathbb{C}$-stable, and that semisimple Lie groups and lattices in semisimple\u0000Lie groups of higher rank are not strictly $mathbb{C}$-stable. Furthermore, we\u0000prove that free groups are not uniformly flexibly $F$-stable over any field\u0000$F$.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a nontrivial finite permutation group of degree $n$. If $G$ is transitive, then a theorem of Jordan states that $G$ has a derangement. Equivalently, a finite group is never the union of conjugates of a proper subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and this can happen even if $G$ has only two orbits, both of which have size $(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture when $G$ acts primitively on at least one of the orbits. Equivalently, we conjecture that a finite group is never the union of conjugates of two proper subgroups of the same order, and we prove this conjecture when at least one of the subgroups is maximal. We prove other cases of the conjecture, and we highlight connections our results have with intersecting families of permutations and roots of polynomials modulo primes. Along the way, we also prove a linear variant on Isbell's conjecture regarding derangements of prime-power order.
{"title":"Orbits of permutation groups with no derangements","authors":"David Ellis, Scott Harper","doi":"arxiv-2408.16064","DOIUrl":"https://doi.org/arxiv-2408.16064","url":null,"abstract":"Let $G$ be a nontrivial finite permutation group of degree $n$. If $G$ is\u0000transitive, then a theorem of Jordan states that $G$ has a derangement.\u0000Equivalently, a finite group is never the union of conjugates of a proper\u0000subgroup. If $G$ is intransitive, then $G$ may fail to have a derangement, and\u0000this can happen even if $G$ has only two orbits, both of which have size\u0000$(1/2+o(1))n$. However, we conjecture that if $G$ has two orbits of size\u0000exactly $n/2$ then $G$ does have a derangement, and we prove this conjecture\u0000when $G$ acts primitively on at least one of the orbits. Equivalently, we\u0000conjecture that a finite group is never the union of conjugates of two proper\u0000subgroups of the same order, and we prove this conjecture when at least one of\u0000the subgroups is maximal. We prove other cases of the conjecture, and we\u0000highlight connections our results have with intersecting families of\u0000permutations and roots of polynomials modulo primes. Along the way, we also\u0000prove a linear variant on Isbell's conjecture regarding derangements of\u0000prime-power order.","PeriodicalId":501037,"journal":{"name":"arXiv - MATH - Group Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142206483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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