{"title":"Residually finite rationally solvable groups and virtual fibring","authors":"Dawid Kielak","doi":"10.1090/jams/936","DOIUrl":"https://doi.org/10.1090/jams/936","url":null,"abstract":"We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79054108","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}
{"title":"The reduction theorem for relatively maximal subgroups","authors":"W. Guo, D. Revin, E. Vdovin","doi":"10.1142/s1664360721500016","DOIUrl":"https://doi.org/10.1142/s1664360721500016","url":null,"abstract":"Let $mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $mathfrak{X}$-maximal in $G/A$. We say that the reduction $mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $mathfrak{X}$-theorem for $A$ implies that $HA/A$ is $mathfrak{X}$-maximal in $G/A$ for every extension $G$ of $A$ and every $mathfrak{X}$-maximal subgroup $H$ of $G$. In this paper, we prove that the reduction $mathfrak{X}$-theorem holds for $A$ if and only if all $mathfrak{X}$-maximal subgroups are conjugate in $A$ and classify the finite groups with this property in terms of composition factors.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82322175","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}
{"title":"Virtual braids and permutations","authors":"P. Bellingeri, L. Paris","doi":"10.5802/aif.3336","DOIUrl":"https://doi.org/10.5802/aif.3336","url":null,"abstract":"Let VB_n be the virtual braid group on n strands and let S_n be the symmetric group on n letters. Let n, m ∈ N such that n ≥ 5, m ≥ 2 and n ≥ m. We determine all possible homomorphisms from VB_n to S_m , from S_n to VB_m and from VB_n to VB_m. As corollaries we get that Out(VB_n) is isomorphic to the Klein group and that VB_n is both Hopfian and co-Hofpian.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79499528","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}
{"title":"Commensurability growth of branch groups","authors":"K. Bou-Rabee, Rachel Skipper, Daniel Studenmund","doi":"10.2140/pjm.2020.304.43","DOIUrl":"https://doi.org/10.2140/pjm.2020.304.43","url":null,"abstract":"Fixing a subgroup $Gamma$ in a group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Delta$ of $G$ with $[Gamma: Gamma cap Delta][Delta : Gamma cap Delta] = n$. For pairs $Gamma leq A$, where $A$ is the automorphism group of a $p$-regular tree and $Gamma$ is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups $Gamma$ (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on $p$-regular trees, this function is precisely $aleph_0$ for any $n = p^k$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"11 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91442105","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}
{"title":"Root systems in number fields","authors":"V. Popov, Y. Zarhin","doi":"10.1512/IUMJ.2021.70.8257","DOIUrl":"https://doi.org/10.1512/IUMJ.2021.70.8257","url":null,"abstract":"We classify the types of root systems $R$ in the rings of integers of number fields $K$ such that the Weyl group $W(R)$ lies in the group $mathcal L(K)$ generated by ${rm Aut} (K)$ and multiplications by the elements of $K^*$. We also classify the Weyl groups of roots systems of rank $n$ which are isomorphic to a subgroup of $mathcal L(K)$ for a number field $K$ of degree $n$ over $mathbb Q$.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"85 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81208141","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}
{"title":"Some results related to finiteness properties of groups for families of subgroups","authors":"Timm von Puttkamer, Xiaolei Wu","doi":"10.2140/agt.2020.20.2885","DOIUrl":"https://doi.org/10.2140/agt.2020.20.2885","url":null,"abstract":"For a group $G$ we consider the classifying space $E_{mathcal{VC}yc}(G)$ for the family of virtually cyclic subgroups. We show that an Artin group admits a finite model for $E_{mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. This solves a conjecture of Juan-Pineda and Leary and a question of L\"uck-Reich-Rognes-Varisco for Artin groups. We then study the conjugacy growth of CAT(0) groups and show that if a CAT(0) group contains a free abelian group of rank two, its conjugacy growth is strictly faster than linear. This also yields an alternative proof for the fact that a CAT(0) cube group admits a finite model for $E_{mathcal{VC}yc}(G)$ if and only if it is virtually cyclic. Our last result deals with the homotopy type of the quotient space $B_{mathcal{VC}yc}(G) = E_{mathcal{VC}yc}(G)/G$. We show for a poly-$mathbb Z$-group $G$, that $B_{mathcal{VC}yc}(G)$ is homotopy equivalent to a finite CW-complex if and only if $G$ is cyclic.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83418320","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}
{"title":"Groups with irreducibly unfaithful subsets for unitary representations","authors":"P. Caprace, P. Harpe","doi":"10.5802/CML.61","DOIUrl":"https://doi.org/10.5802/CML.61","url":null,"abstract":"Let $G$ be a group. A subset $F subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) ne text{id}$ for all $x in F smallsetminus {e}$. Otherwise $F$ is called irreducibly unfaithful. Given a positive integer $n$, we say that $G$ has Property $P(n)$ if every subset of size $n$ is irreducibly faithful. Every group has $P(1)$, by a classical result of Gelfand and Raikov. Walter proved that every group has $P(2)$. It is easy to see that some groups do not have $P(3)$. \u0000We provide a complete description of the irreducibly unfaithful subsets of size $n$ in a (finite or infinite) countable group $G$ with Property $P(n-1)$: it turns out that such a subset is contained in a finite elementary abelian normal subgroup of $G$ of a particular kind. We deduce a characterization of Property $P(n)$ purely in terms of the group structure. It follows that, if a countable group $G$ has $P(n-1)$ and does not have $P(n)$, then $n$ is the cardinality of a projective space over a finite field. \u0000A group $G$ has Property $Q(n)$ if, for every subset $F subset G$ of size at most $n$, there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) ne pi(y)$ for any distinct $x, y$ in $F$. Every group has $Q(2)$. For countable groups, it is shown that Property $Q(3)$ is equivalent to $P(3)$, Property $Q(4)$ to $P(6)$, and Property $Q(5)$ to $P(9)$. For $m, n ge 4$, the relation between Properties $P(m)$ and $Q(n)$ is closely related to a well-documented open problem in additive combinatorics.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90523499","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}
{"title":"Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks","authors":"S. Francaviglia, A. Martino","doi":"10.1090/tran/8333","DOIUrl":"https://doi.org/10.1090/tran/8333","url":null,"abstract":"This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. \u0000In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90984786","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}
{"title":"Displacements of automorphisms of free groups II: Connectivity of level sets and decision problems","authors":"S. Francaviglia, A. Martino","doi":"10.1090/tran/8535","DOIUrl":"https://doi.org/10.1090/tran/8535","url":null,"abstract":"This is the second of two papers in which we investigate the properties of displacement functions of automorphisms of free groups (more generally, free products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial bordification. We develop a theory for both reducible and irreducible autormorphisms. As we reach the bordification of $CV_n$ we have to deal with general deformation spaces, for this reason we developed the theory in such generality. In first paper~cite{FMpartI} we studied general properties of the displacement functions, such as well-orderability of the spectrum and the topological characterization of min-points via partial train tracks (possibly at infinity). This paper is devoted to proving that for any automorphism (reducible or not) any level set of the displacement function is connected. As an application, this result provides a stopping procedure for brute force search algorithms in $CV_n$. We use this to reprove two known algorithmic results: the conjugacy problem for irreducible automorphisms and detecting irreducibility of automorphisms. Note: the two papers were originally packed together in the preprint arxiv:1703.09945. We decided to split that paper following the recommendations of a referee.","PeriodicalId":8427,"journal":{"name":"arXiv: Group Theory","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88050586","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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