Subgroup Of Finite Index Research Articles

Profinite Rigidity, Kleinian Groups, and the Cofinite Hopf Property

Let Γ be a nonelementary Kleinian group and H<Γ be a finitely generated, proper subgroup. We prove that if Γ has finite covolume, then the profinite completions of H and Γ are not isomorphic. If H has finite index in Γ, then there is a finite group onto which H maps but Γ does not. These results streamline the existing proofs that there exist full-sized groups that are profinitely rigid in the absolute sense. They build on a circle of ideas that can be used to distinguish among the profinite completions of subgroups of finite index in other contexts, for example, limit groups. We construct new examples of profinitely rigid groups, including the fundamental group of the hyperbolic 3-manifold Vol(3) and that of the 4-fold cyclic branched cover of the figure-eight knot. We also prove that if a lattice in PSL(2,C) is profinitely rigid, then so is its normalizer in PSL(2,C).

GROUPS WITH A LARGE PERMUTABLY EMBEDDED SUBGROUP

AbstractA group is called quasihamiltonian if all its subgroups are permutable, and we say that a subgroup Q of a group G is permutably embedded in G if $\langle Q,g\rangle $ is quasihamiltonian for each element g of G. It is proved here that if a group G contains a permutably embedded normal subgroup Q such that $G/Q$ is Černikov, then G has a quasihamiltonian subgroup of finite index; moreover, if G is periodic, then it contains a Černikov normal subgroup N such that $G/N$ is quasihamiltonian. This result should be compared with theorems of Černikov and Schlette stating that if a group G is Černikov over its centre, then G is abelian-by-finite and its commutator subgroup is Černikov.

On groups in which every element has a prime power order and which satisfy some boundedness condition

In this paper, we shall deal with periodic groups, in which each element has a prime power order. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each [Formula: see text]-element of [Formula: see text] is of order [Formula: see text]. A group [Formula: see text] will be called a [Formula: see text]-group if each element of [Formula: see text] has a prime power order and for each [Formula: see text] there exists a positive integer [Formula: see text] such that each finite [Formula: see text]-subgroup of [Formula: see text] is of order [Formula: see text]. Here, [Formula: see text] denotes the set of all primes dividing the order of some element of [Formula: see text]. Our main results are the following four theorems. Theorem 1: Let [Formula: see text] be a finitely generated [Formula: see text]-group. Then [Formula: see text] has only a finite number of normal subgroups of finite index. Theorem 4: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a locally finite group. Theorem 7: Let [Formula: see text] be a locally graded [Formula: see text]-group. Then [Formula: see text] is a finite group. Theorem 8: Let [Formula: see text] be a [Formula: see text]-group satisfying [Formula: see text]. Then [Formula: see text] is a locally finite group.

Unramified correspondences and virtual properties of mapping class groups

We establish a connection between the conjecture of Bogomolov–Tschinkel about unramified correspondences and the Ivanov conjecture about the virtual homology of mapping class groups. Given g ⩾ 2 $g\geqslant 2$ , we show that every genus g $g$ Riemann surface X $X$ virtually dominates a fixed Riemann surface Y $Y$ of genus at least two if and only if there exists a finite index subgroup Γ < Mod g 1 $\Gamma <\operatorname{Mod}^1_g$ which allows a point pushing epimorphism onto a free group of rank two. As a consequence of this result, we show that the Putman–Wieland conjecture about the Higher Prym representations does not hold when g = 2 $g=2$ .

C1 actions on the circle of finite index subgroups of Mod(Σg), Aut(Fn), and Out(Fn)

Let Σg be a closed, connected, and oriented surface of genus g≥24, and let Γ be a finite index subgroup of the mapping class group Mod(Σg) that contains the Torelli group ℐ(Σg). Then any orientation-preserving C1 action of Γ on the circle cannot be faithful. We also show that if Γ is a finite index subgroup of Aut(Fn), when n≥8, that contains the subgroup of IA-automorphisms, then any orientation-preserving C1 action of Γ on the circle cannot be faithful. Similarly, if Γ is a finite index subgroup of Out(Fn), when n≥8, that contains the Torelli group 𝒯n, then any orientation preserving C1 action of Γ on the circle cannot be faithful. In fact, when n≥10, any orientation-preserving C1 action of a finite index subgroup of Aut(Fn) or Out(Fn) on the circle cannot be faithful.

Level structure, arithmetic representations, and noncommutative Siegel linearization

Abstract Let ℓ {\ell} be a prime, k a finitely generated field of characteristic different from ℓ {\ell} , and X a smooth geometrically connected curve over k. Say a semisimple representation of π 1 ét ⁢ ( X k ¯ ) {\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} is arithmetic if it extends to a finite index subgroup of π 1 ét ⁢ ( X ) {\pi_{1}^{{\text{\'{e}t}}}(X)} . We show that there exists an effective constant N = N ⁢ ( X , ℓ ) {N=N(X,\ell)} such that any semisimple arithmetic representation of π 1 ét ⁢ ( X k ¯ ) {\pi_{1}^{{\text{\'{e}t}}}(X_{\bar{k}})} into GL n ⁡ ( ℤ ℓ ¯ ) {\operatorname{GL}_{n}(\overline{\mathbb{Z}_{\ell}})} , which is trivial mod ℓ N {\ell^{N}} , is in fact trivial. This extends a previous result of the second author from characteristic zero to all characteristics. The proof relies on a new noncommutative version of Siegel’s linearization theorem and the ℓ {\ell} -adic form of Baker’s theorem on linear forms in logarithms.

Criterion of proper actions of abelian affine discontinuous groups for ℝn

In [Formula: see text], Auslander conjectured that every crystallographic subgroup [Formula: see text] of the affine group [Formula: see text] acting properly discontinuously on [Formula: see text] is virtually solvable, i.e. contains a solvable subgroup of finite index. One of the major difficulties to prove such a conjecture is to generate a criterion of the proper action of a given discrete group acting properly discontinuously on [Formula: see text]. In this paper, we focus on this question in the setting where [Formula: see text] is abelian. We show that [Formula: see text] acts properly on [Formula: see text] if and only if any [Formula: see text]-orbit is discrete (or equivalelntly closed). We further generate an equivalent criterion in the setting where [Formula: see text] stands for a crystallographic discontinuous affine group for [Formula: see text] based upon a generating family.

Central units of integral group rings of monomial groups

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring Z G \mathbb {Z}G for a subgroup closed monomial group G G with the property that every cyclic subgroup of order not a divisor of 4 4 or 6 6 is subnormal in G G . If G G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) . Furthermore, for a generalized strongly monomial group G G , the rank of Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G G .

Generating ideals by additive subgroups of rings

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings.Let R be any ring equipped with an arbitrary additional first order structure, and A a set of parameters. We show that whenever H is an A-definable, finite index subgroup of (R,+), then H+R⋅H contains an A-definable, two-sided ideal of finite index. As a corollary, we obtain a positive answer to Question 4.9 of [4]: if R is unital, then (R¯,+)A00+R¯⋅(R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00, which also implies that R¯A00=R¯A000, where R¯≻R is a sufficiently saturated elementary extension of R, (R¯,+)A00 [resp. R¯A00] denotes the smallest A-type-definable, bounded index additive subgroup [resp. ideal] of R¯, and R¯A000 is the smallest invariant over A, bounded index ideal of R¯. If R is of positive characteristic (not necessarily unital), we get a sharper result: (R¯,+)A00+R¯⋅(R¯,+)A00=R¯A00. We obtain a similar result (but with more steps required) for finitely generated (not necessarily unital) rings. We obtain analogous results for topological rings. The above result for unital rings implies that the simplified descriptions of the definable (and so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 4.5 of [4] are valid for all unital rings.We also analyze many concrete examples, where we compute the number of steps needed to generate a group by (R¯∪{1})⋅(R¯,+)A00 and study related aspects, showing “optimality” of some of our main results and yielding answers to some natural questions.

Virtual Critical Regularity of Mapping Class Group Actions on the Circle

We show that if G1 and G2 are non-solvable groups, then no C1,τ action of \((G_{1}\times G_{2})*\mathbb {Z}\) on S1 is faithful for τ > 0. As a corollary, if S is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group Mod(S) with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.

Automorphic Lie Algebras and Modular Forms

AbstractWe introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma $ be a finite index subgroup of $\textrm {SL}(2,\mathbb Z)$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\textrm {SL}(2,{\mathbb {C}})$. We show that the Lie algebra of the corresponding $\mathfrak {g}$-valued modular forms is isomorphic to the extension of $\mathfrak {g}$ over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups $\Gamma (N), \, N\leq 6$, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras.

Invariable generation and the Houghton groups

The Houghton groups H1,H2,… are a family of infinite groups. In 1977 Wiegold showed that H3 was invariably generated (IG) but H1⩽H3 was not. A natural question is then whether the groups H2,H3,… are all IG. Wiegold also ends by saying that, in the examples he had found of an IG group with a subgroup that is not IG, the subgroup was never of finite index. Another natural question is then whether there is a subgroup of finite index in H3 that is not IG. In this note we prove, for each n∈{2,3,…}, that Hn and all of its finite index subgroups are IG.The independent work of Minasyan and Goffer-Lazarovich in June 2020 frames this note quite nicely: they showed that an IG group can have a finite index subgroup that is not IG.

On the spread of infinite groups

AbstractA group is $\frac 32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$. We then show that $G_1$ and $G_2$ are $\frac 32$-generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$, we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$-generated whereas $G_{2k}$ is not.

Bounds on degrees of covers with injective monodromy in iterated Kodaira fibrations

Let π:X→Y be an n-dimensional iterated Kodaira fibration with fiber of genus g and injective monodromy. In [6], it is proved that we can pass to a finite index subgroup of π1(X) to get the base space of an (n+1)-dimensional iterated Kodaira fibration with injective monodromy, and they asked about bounding the index of such a group. We provide a bound on this index.

Meromorphic modular forms and the three-loop equal-mass banana integral

We consider a class of differential equations for multi-loop Feynman integrals which can be solved to all orders in dimensional regularisation in terms of iterated integrals of meromorphic modular forms. We show that the subgroup under which the modular forms transform can naturally be identified with the monodromy group of a certain second-order differential operator. We provide an explicit decomposition of the spaces of modular forms into a direct sum of total derivatives and a basis of modular forms that cannot be written as derivatives of other functions, thereby generalising a result by one of the authors form the full modular group to arbitrary finite-index subgroups of genus zero. Finally, we apply our results to the two- and three-loop equal-mass banana integrals, and we obtain in particular for the first time complete analytic results for the higher orders in dimensional regularisation for the three-loop case, which involves iterated integrals of meromorphic modular forms.

On the finite index subgroups of Houghton\u2019s groups

Houghton’s groups H_2, H_3, ldots are certain infinite permutation groups acting on a countably infinite set; they have been studied, among other things, for their finiteness properties. In this note we describe all of the finite index subgroups of each Houghton group, and their isomorphism types. Using the standard notation that d(G) denotes the minimal size of a generating set for G, we then show, for each nin {2, 3,ldots } and U of finite index in H_n, that d(U)in {d(H_n), d(H_n)+1} and characterise when each of these cases occurs.

On the Nielsen-Schreier Theorem in Homotopy Type Theory

We give a formulation of the Nielsen-Schreier theorem (subgroups of free groups are free) in homotopy type theory using the presentation of groups as pointed connected 1-truncated types. We show the special case of finite index subgroups holds constructively and the full theorem follows from the axiom of choice. We give an example of a boolean infinity topos where our formulation of the theorem does not hold and show a stronger "untruncated" version of the theorem is provably false in homotopy type theory.

On Groups in Which Many Automorphisms Are Cyclic

Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup ⟨x,xα⟩ is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G/Z(G). In this paper, the class FW of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated.

The Ribes-Zalesskii property of some one relator groups

The profinite topology on any abstract group $G$, is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group $G$ has the Ribes-Zalesskii property of rank $k$, or is RZ$_{k}$ with $k$ a natural number, if any product $H_{1} H_{2} \cdots H_{k}$ of finitely generated subgroups $H_{1}, H_{2}, \cdots , H_{k}$ is closed in the profinite topology on $G$. And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ$_{k}$ for any natural number $k$. In this paper we characterize groups which are RZ$_{2}$. Consequently, we obtain condition under which a free product with amalgamation of two RZ$_{2}$ groups is RZ$_{2}$. After observing that the Baumslag-Solitar groups $BS (m, n)$ are RZ$_{2}$ and clearly RZ if $m= n$, we establish some suitable properties on the RZ$_{2}$ property for the case when $m= -n$. Finally, since any group $BS (m, n)$ can be viewed as a HNN-extension, then we point out the Ribes-Zalesskii property of rank two on some HNN-extensions.

Cantor dynamics of renormalizable groups

A group \Gamma is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding \varphi \colon \Gamma \to \Gamma whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for \Gamma .” In this work, we associate a dynamical system to a renormalization \varphi of \Gamma . The discriminant invariant {\mathcal D}_{\varphi} of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If {\mathcal D}_{\varphi} is a finite group for some renormalization, we show that \Gamma/C_{\varphi} is virtually nilpotent, where C_{\varphi} is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.