We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We extend work of Berdinsky and Khoussainov [‘Cayley automatic representations of wreath products’, International Journal of Foundations of Computer Science27(2) (2016), 147–159] to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.
The paper gives a simple proof of Graev’s theorem (asserting that the free product of Hausdorff topological groups is Hausdorff) for a particular case which includes the countable case of
$k_\omega $-groups and the countable case of Lindelöf P-groups. For this it is shown that in these particular cases the topology of the free product of Hausdorff topological groups coincides with the
$X_0$-topology in the Mal’cev sense, where X is the disjoint union of the topological groups identifying their units.
We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.
Let $G(n)={\textrm {Sp}}(n,1)$ or
${\textrm {SU}}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in
$G(n)$. Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for
${\textrm {SU}}(3,1)$. We extend this notion and classify
$G(n)$-conjugation orbits of such elements in arbitrary dimension. For
$n=3$, they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus
$g \geq 2$) oriented surface into
$G(3)$.
Soient K un corps discrètement valué et hensélien,
${\mathcal {O}}$ son anneau d'entiers supposé excellent,
$\kappa $ son corps résiduel supposé parfait et G un K-groupe quasi-réductif, c'est-à-dire lisse, affine, connexe et à radical unipotent déployé trivial. On construit l'immeuble de Bruhat-Tits
${\mathcal {I}}(G, K)$ pour
$G(K)$ de façon canonique, améliorant les constructions moins canoniques de M. Solleveld sur les corps locaux, et l'on associe un
${\mathcal {O}}$-modèle en groupes
${\mathcal {G}}_{\Omega }$ de G à chaque partie non vide et bornée
$\Omega $ contenue dans un appartement de
${\mathcal {I}}(G,K)$. On montre que les groupes parahoriques
${\mathcal {G}}_{\textbf {f}}$ attachés aux facettes peuvent être caractérisés en fonction de la géométrie de leurs grassmanniennes affines, ainsi que dans la thèse de T. Richarz. Ces résultats sont appliqués ailleurs à l'étude des grassmanniennes affines tordues entières.
The pronorm of a group G is the set
$P(G)$ of all elements
$g\in G$ such that X and
$X^g$ are conjugate in
${\langle {X,X^g}\rangle }$ for every subgroup X of G. In general the pronorm is not a subgroup, but we give evidence of some classes of groups in which this property holds. We also investigate the structure of a generalised soluble group G whose pronorm contains a subgroup of finite index.
Parapolar spaces are point-line geometries introduced as a geometric approach to (exceptional) algebraic groups. We characterize a wide class of Lie geometries as parapolar spaces satisfying a simple intersection property. In particular, many of the exceptional Lie incidence geometries occur. In an appendix, we extend our result to the locally disconnected case and discuss the locally disconnected case of some other well-known characterizations.
A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$. (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$.) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.
Let
$a_1$
,
$a_2$
, and
$a_3$
be distinct reduced residues modulo q satisfying the congruences
$a_1^2 \equiv a_2^2 \equiv a_3^2 \ (\mathrm{mod}\ q)$
. We conditionally derive an asymptotic formula, with an error term that has a power savings in q, for the logarithmic density of the set of real numbers x for which
$\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)$
. The relationship among the
$a_i$
allows us to normalize the error terms for the
$\pi (x;q,a_i)$
in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.
Ryabukhin showed that there is a correspondence between elementary radical classes of rings and certain filters of ideals of the free ring on one generator, analogous to the Gabriel correspondence between torsion classes of left unital modules and certain filters of left ideals of the coefficient ring. This correspondence is further explored here. All possibilities for the intersection of the ideals in a filter are catalogued, and the connections between filters and other ways of describing elementary radical classes are investigated. Some generalisations to nonassociative rings and groups are also presented.
Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.
If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.
It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.
To every dynamical system
$(X,\varphi )$
over a totally disconnected compact space, we associate a left-orderable group
$T(\varphi )$
. It is defined as a group of homeomorphisms of the suspension of
$(X,\varphi )$
which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system
$(X,\varphi )$
, the group
$T(\varphi )$
does not have infinite subgroups with Kazhdan's property
$(T)$
. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if
$(X,\varphi )$
has a dense orbit, then the isomorphism type of the group
$T(\varphi )$
is a complete invariant of flow equivalence of the pair
$\{\varphi , \varphi ^{-1}\}$
.
Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.
In 1945–1946, C. L. Siegel proved that an $n$-dimensional lattice $\unicode[STIX]{x1D6EC}$ of determinant $\text{det}(\unicode[STIX]{x1D6EC})$ has at most $m^{n^{2}}$ different sublattices of determinant $m\cdot \text{det}(\unicode[STIX]{x1D6EC})$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. We present a systematic treatment for counting the sublattices and derive a formula for the number of the sublattice classes under unimodular equivalence.
We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.
As an application, we provide a short proof (modulo Bass–Serre theory) of Vinogradov’s result that the free product of left-orderable groups is left-orderable.
We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$, where $C$ is a finite group and $F$ a non-abelian free group.
We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in $\unicode[STIX]{x1D6E4}$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.