Subgroup Of Finite Index Research Articles

Units of twisted group rings and their correlations to classical group rings

This paper is centred around the classical problem of extracting properties of a finite group G from the ring isomorphism class of its integral group ring ZG. This problem is considered via describing the unit group U(ZG) generically for a finite group. Since the ‘90s’ several well known generic constructions of units are known to generate a subgroup of finite index in U(ZG) if QG does not have so-called exceptional simple epimorphic images, e.g. M2(Q). However it remained a major open problem to find a generic construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation U(ZG)ab. To obtain all this, we investigate the group ring RΓ of an extension Γ of some normal subgroup N by a group G, over a domain R. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of Γ over the fraction field F of R in terms of various twisted group rings of G over finite extensions of F. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of U(RΓ) via twisted group rings.

On groups whose cogrowth series is the diagonal of a rational series

We show that if a group contains [Formula: see text] as a finite-index subgroup, then its cogrowth series is the diagonal of a rational function for every generating set. This answers a question of Pak and Soukup on the cogrowth of virtually abelian groups; and generalizes a result by Elder, Rechnitzer, Janse van Rensburg, and Wong on the cogrowth series of the Baumslag–Solitar groups [Formula: see text].

Rigidity of the Torelli subgroup in $\mathrm{Out}(F_{N})$

Let N\ge 4 . We prove that every injective homomorphism from the Torelli subgroup \textup{IA}_{N} to \textup{Out}(F_{N}) differs from the inclusion by a conjugation in \textup{Out}(F_{N}) . This applies more generally to the following subgroups of \textup{Out}(F_{N}) : every finite-index subgroup of \textup{Out}(F_{N}) (recovering a theorem of Farb and Handel); every subgroup of \textup{Out}(F_{N}) that contains a finite-index subgroup of one of the groups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) ; every subgroup that contains a power of every linearly-growing automorphism; more generally, every twist-rich subgroup of \textup{Out}(F_{N}) –those are subgroups that contain sufficiently many twists in an appropriate sense.Among applications, this recovers the fact that the abstract commensurator of every group above is equal to its relative commensurator in \textup{Out}(F_{N}) ; it also implies that all subgroups in the Andreadakis–Johnson filtration of \textup{Out}(F_{N}) are co-Hopfian.We also prove the same rigidity statement for subgroups of \textup{Out}(F_{3}) which contain a power of every Nielsen transformation. This shows, in particular, that \textup{Out}(F_{3}) and all its finite-index subgroups are co-Hopfian, extending a theorem of Farb and Handel to the N=3 case.

Centralisers of linear growth automorphisms of free groups

Abstract In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

The action of [formula omitted]-shadows on child's drawings

GT-shadows [8] are tantalizing objects that can be thought of as approximations of elements of the mysterious Grothendieck-Teichmueller group GTˆ introduced by V. Drinfeld in 1990. GT-shadows form a groupoid GTSh whose objects are finite index subgroups of the pure braid group PB4, that are normal in B4. The goal of this paper is to describe the action of GT-shadows on Grothendieck's child's drawings and show that this action agrees with that of GTˆ. We discuss the hierarchy of orbits of child's drawings with respect to the actions of GTSh, GTˆ, and the absolute Galois group GQ of rationals. We prove that the monodromy group and the passport of a child's drawing are invariant with respect to the action of the subgroupoid GTSh♡ of charming GT-shadows. We use the action of GT-shadows on child's drawings to prove that every Abelian child's drawing admits a Belyi pair defined over Q. Finally, we describe selected examples of non-Abelian child's drawings.

ADJUNCTION OF ROOTS TO PROFINITE GROUPS

In this paper, when $G'$ is a group obtained by adjoining a $n$th-root of $g$ to a given group $G$, where $n$ is a nonzero natural number and $g$ is an element of $G$ of infinite order, we compute the profinite completion $\widehat{G'}$ of $G'$. Also, given $G$ a profinite group in which any subgroup of finite index is open, $n$ a nonzero natural number, $g$ an element of $G$, and $x$ an element not belonging to $G$, we point out necessary and sufficient condition under which the group obtained by adjoining roots to the profinite group $G$ remains again profinite. Our proofs make use of theoretico-combinatorial methods.

Profinite non-rigidity of arithmetic groups

We show that for a typical higher-rank arithmetic lattice \Gamma , there exist finite index subgroups \Gamma_{1} and \Gamma_{2} such that \Gamma_{1}\not\cong \Gamma_{2} while \widehat{\Gamma_{1}}\cong \widehat{\Gamma_{2}} . But there are exceptions to that rule.

Virtual endomorphisms of the group $pg$

A virtual endomorphism of a group $G$ is a homomorphism of the form $\phi:H\rightarrow G$, where $H<G$ is a subgroup of finite index. A virtual endomorphism $\phi:H\rightarrow G$ is called simple if there are no nontrivial normal $\phi$-invariant subgroups, that is, the $\phi$-core is trivial. We describe all virtual endomorphisms of the plane group $pg$, also known as the fundamental group of the Klein bottle. We determine which of these virtual endomorphisms are simple, and apply these results to the self-similar actions of the group. We prove that the group $pg$ admits a transitive self-similar (as well as finite-state) action of degree $d$ if and only if $d\geq 2$ is not an odd prime, and admits a self-replicating action of degree $d$ if and only if $d\geq 6$ is not a prime or a power of $2$.

Multiplier systems for Siegel modular groups

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

Estimates of Picard modular cusp forms

Abstract In this article, for n ≥ 2 {n\geq 2} , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) {\mathrm{SU}((n,1),\mathbb{C})} . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) {\Gamma\subset\mathrm{SU}((2,1),\mathcal{O}_{K})} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k {{{\mathcal{B}_{\Gamma}^{k}}}} denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 {\mathbb{B}^{2}} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 {X_{\Gamma}:=\Gamma\backslash\mathbb{B}^{2}} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet {|\cdot|_{\mathrm{pet}}} denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) {\mathcal{S}_{k}(\Gamma)} . Then, for k ≫ 1 {k\gg 1} , we have the following estimate: sup z ∈ X Γ ⁡ | ℬ Γ k ⁢ ( z ) | pet = O Γ ⁢ ( k 5 2 ) , \sup_{z\in X_{\Gamma}}|{{\mathcal{B}_{\Gamma}^{k}}}(z)|_{\mathrm{pet}}=O_{% \Gamma}(k^{\frac{5}{2}}), where the implied constant depends only on Γ.

On computing finite index subgroups of PSL2(ℤ)

We present a method to compute finite index subgroups of PSL2(ℤ). Our strategy follows Kulkarni’s ideas, the main contribution being a recursive method to compute bivalent trees as well as their automorphism groups. As a concrete application, we compute all subgroups of index up to [Formula: see text]. We then use this database to produce tables with several arithmetical properties.

On virtual indicability and property (T) for outer automorphism groups of RAAGs

We give a condition on the defining graph of a right-angled Artin group, which implies that its automorphism group is virtually indicable, that is, it has a finite index subgroup that admits a homomorphism onto \mathbb{Z} . We use this as a part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence, we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.

A generalization of the Davis–Moussong complex for Dyer groups

A common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. Matthew Dyer introduced a class of groups, which we call Dyer groups, sharing this feature. This class includes, but is not limited to, Coxeter groups, right-angled Artin groups, and graph products of cyclic groups. We introduce Dyer groups by giving their standard presentation and show that they are finite-index subgroups of Coxeter groups. We then introduce a piecewise Euclidean cell complex \Sigma which generalizes the Davis–Moussong complex and the Salvetti complex. The construction of \Sigma uses simple categories without loops and complexes of groups. We conclude by proving that the cell complex \Sigma is \mathrm{CAT}(0) .

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.

Failure of the well-rounded retract for outer space and Teichmüller space

The well-rounded retract for S L n ( Z ) SL_n(\mathbb {Z}) is defined as the set of flat tori of unit volume and dimension n n whose systoles generate a finite-index subgroup in homology. This set forms an equivariant spine of minimal dimension for the space of flat tori. For both the Outer space X n X_n of metric graphs of rank n n and the Teichmüller space T g \mathcal {T}_g of closed hyperbolic surfaces of genus g g , we show that the literal analogue of the well-rounded retract does not contain an equivariant spine. We also prove that the sets of graphs whose systoles fill either topologically or geometrically (two analogues of a set proposed as a spine for T g \mathcal {T}_g by Thurston) are spines for X n X_n but that their dimension is larger than the virtual cohomological dimension of O u t ( F n ) Out(F_n) in general.

Colorings of patterns fixed by an arbitrary finite-index subgroup of the symmetry group.

In this research project, a framework is established that characterizes the colorings of a pattern fixed by an arbitrarily chosen finite-index subgroup of the pattern's symmetry group. This framework is applied to the sphalerite structure to enumerate all colorings fixed by two different subgroups of the symmetry group. These colorings are used to perform ordered substitution on the atoms in the sphalerite structure and demonstrate some of the well known derived structures (e.g. chalcopyrite, stannite). Some results involving equivalences of colorings fixed by the chosen subgroup are also developed.

Algebraic and context-free subsets of subgroups

We study the relation between the structure of algebraic and context-free subsets of a group G and that of a finite index subgroup H. Using these results, we prove that a kind of Fatou property, previously studied by Berstel and Sakarovitch in the context of rational subsets and by Herbst in the context of algebraic subsets, holds for context-free subsets if and only if the group is virtually free. We also exhibit a counterexample to a question of Herbst concerning this property for algebraic subsets.

Correction to: On the finite index subgroups of Houghton’s groups

This note resolves an issue raised by Prof. Derek Holt for the paper “On the finite index subgroups of Houghton’s groups”.

Finite index subgroups in Chevalley groups are bounded: An addendum to “On Bi-Invariant Word Metrics”

We prove that finite index subgroups in [Formula: see text]-arithmetic Chevalley groups are bounded.

Products of subgroups, subnormality, and relative orders of elements

Let G be a group. We give an explicit description of the set of elements x ∈ G such that x|G:H| ∈ H for every subgroup of finite index H ≤ G. This is related to the following problem: given two subgroups H and K, with H of finite index, when does |HK:H| divide |G:H|?