Finite Index Research Articles

The twisted derivation problem for group rings

We study $$(\sigma ,\tau )$$ -derivations of a group ring RG where G is a group with center having finite index in G and R is a semiprime ring with 1 such that either R has no torsion elements or that if R has p-torsion elements, then p does not divide the order of G and let $$\sigma ,\tau $$ be R-linear endomorphisms of RG fixing the center of RG pointwise. We generalize Main Theorem 1.1 of Chaudhuri (Comm Algebra 47(9): 3800–3807, 2019) and prove that there is a ring $$T\supset R$$ such that $$\mathcal {Z}(T)\supset \mathcal {Z}(R)$$ and that for the natural extensions of $$\sigma , \tau $$ to TG, we get $$H^1(TG,{}_\sigma TG_\tau )=0$$ , where $${}_\sigma TG_\tau $$ is the twisted $$TG-TG$$ -bimodule. We provide applications of the above result and Main Theorem 1.1 of Chaudhuri (2019) to integral group rings of finite groups and connect twisted derivations of integral group rings to other important problems in the field such as the isomorphism problem and the Zassenhaus conjectures. We also give an example of a group G which is both locally finite and nilpotent and such that for every field F, there exists an F-linear $$\sigma $$ -derivation of FG which is not $$\sigma $$ -inner.

On a composition of subfactors with group subfactors

Let M be a type II1 factor, G be a finite group, and N ⊆ M be an irreducible subfactor of finite index. We prove that the composed lattice of the intermediate subfactor lattice for the inclusion N ⊆ M and the subgroup lattice of G can be realized as an intermediate subfactor lattice of a certain composed subfactor of finite index, and this subfactor also has finite depth when N ⊆ M has finite depth.

Recent progress on stable and finite Morse index solutions of semilinear elliptic equations

<p style='text-indent:20px;'>We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.</p>

The Birman exact sequence does not virtually split

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $\Gamma$ contained in the Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact sequence for any finite-index subgroup of the full mapping class group. This theorem was announced in a 1990 preprint of G. Mess, but an error was uncovered and described in a recent paper of the first author.

Johnson pseudo-Connes amenability of dual Banach algebras

We introduce the notion of Johnson pseudo-Connes amenability for dual Banach algebras. We study the relation between this new notion with the various notions of Connes amenability like Connes amenability, approximate Connes amenability and pseudo Connes amenability. We also investigate some hereditary properties of this new notion. We prove that for a locally compact group G,M(G) is Johnson pseudo-Connes amenable if and only if G is amenable. Also we show that for every non-empty set I,MI(C) under this new notion is forced to have a finite index. Finally, we provide some examples of certain dual Banach algebras and we study their Johnson pseudo-Connes amenability.

Hilbert boundary value problem for generalized analytic functions with a singular line

In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.

Counting finite index subrings of $\mathbb Z^n$

We count subrings of small index of $\mathbb {Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers $k$ that

Cusp and b_1 growth for ball quotients and maps onto Z with finitely generated kernel

Let $M = \mathbb{B}^2 / \Gamma$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a lattice in $\mathrm{PU}(2, 1)$ admitting a homomorphism onto $\mathbb{Z}$ with finitely generated kernel. Moreover, we show that any cocompact arithmetic lattice $\Gamma \subset \mathrm{PU}(n, 1)$ of simplest type contains a finite index subgroup with this property.

On a bottom layer in a group

We consider the problem of recognizing a group by its bottom layer. This problem is solved in the class of layer-finite groups. A group is layer-finite if it has a finite number of elements of every order. This concept was first introduced by S. N. Chernikov. It appeared in connection with the study of infinite locally finite p-groups in the case when the center of the group has a finite index. S. N. Chernikov described the structure of an arbitrary group in which there are only finite elements of each order and introduced the concept of layer-finite groups in 1948. Bottom layer of the group G is a set of its elements of prime order. If have information about the bottom layer of a group we can receive results about its recognizability by bottom layer. The paper presents the examples of groups that are recognizable, almost recognizable and unrecognizable by its bottom layer under additional conditions.

Torsion-free abelian groups revisited

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut( G ) on the set of maximal independent subsets of G determine the indecomposable decompositions. G contains a direct sum of pure strongly indecomposable groups as a subgroup of finite index.

Lower central series of Baumslag–Solitar groups

We find the lower central series for residually nilpotent Baumslag-Solitar groups, and find the intersection of all terms of the lower central series. Also, we find non-abelian Bauslag-Solitar groups for which the lower central series has length 2. For some Baumslag-Solitar groups a connection is found between the intersection of all subgroups of finite index and the intersection of all terms of the lower central series.

On stable and finite Morse index solutions of the fractional Toda system

We develop a monotonicity formula for solutions of the fractional Toda system(−Δ)sfα=e−(fα+1−fα)−e−(fα−fα−1)inRn, when 0<s<1, α=1,⋯,Q, f0=−∞, fQ+1=∞ and Q≥2 is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when n>2s andΓ(n2)Γ(1+s)Γ(n−2s2)Q(Q−1)2>Γ2(n+2s4)Γ2(n−2s4). Here, Γ is the Gamma function. When Q=2, the above equation is the classical (fractional) Gelfand-Liouville equation.

Groups with a solvable subgroup of prime-power index

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index $p^{\alpha}$ (for some prime $p$), then the quotient $G/\mbox{rad}(G)$ of $G$ over the solvable radical is asymptotically small in comparison to $p^{\alpha}!$ (Theorem 4).

Residually finite dimensional algebras and polynomial almost identities

Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,

Finite index theorems for iterated Galois groups of unicritical polynomials

Let K K be the function field of a smooth irreducible curve defined over Q ¯ \overline {Q} . Let f ∈ K [ x ] f\in K[x] be of the form f ( x ) = x q + c f(x)=x^q+c , where q = p r , r ≥ 1 , q = p^{r}, r \ge 1, is a power of the prime number p p , and let β ∈ K ¯ \beta \in \overline {K} . For all n ∈ N ∪ { ∞ } n\in \mathbb {N}\cup \{\infty \} , the Galois groups G n ( β ) = G a l ( K ( f − n ( β ) ) / K ( β ) ) G_n(\beta )=\mathrm {Gal}(K(f^{-n}(\beta ))/K(\beta )) embed into [ C q ] n [C_q]^n , the n n -fold wreath product of the cyclic group C q C_q . We show that if f f is not isotrivial, then [ [ C q ] ∞ : G ∞ ( β ) ] &gt; ∞ [[C_q]^\infty :G_\infty (\beta )]&gt;\infty unless β \beta is postcritical or periodic. We are also able to prove that if f 1 ( x ) = x q + c 1 f_1(x)=x^q+c_1 and f 2 ( x ) = x q + c 2 f_2(x)=x^q+c_2 are two such distinct polynomials, then the fields ⋃ n = 1 ∞ K ( f 1 − n ( β ) ) \bigcup _{n=1}^\infty K(f_1^{-n}(\beta )) and ⋃ n = 1 ∞ K ( f 2 − n ( β ) ) \bigcup _{n=1}^\infty K(f_2^{-n}(\beta )) are disjoint over a finite extension of K K .

Double Affine Hecke Algebras and Congruence Groups

The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type $A_{2}$, which descends to a faithful outer action of a congruence subgroup of $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$. This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. The structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products $W\ltimes Q^{\vee}$ of the Weyl group $W$ with the coroot lattice $Q^{\vee}$. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type $A_{2},B_{2},G_{2}$ act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist $r=1,2,3$, respectively. This extends a fundamental result of Cherednik for $r=1$. We show further that the above rank two Artin group action descends to an outer action of congruence subgroup $\Gamma_{1}(r)$. In particular $\Gamma_{1}( r) $ acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist type $r$, giving rise to a projective representation of $\Gamma_{1}( r) $ on the space of a $\Gamma_{1}( r) $-stable representation.

English

Let H denote a finite index subgroup of the modular group Γ and let ϱ denote a finite-dimensional complex representation of H. Let M(ϱ) denote the collection of holomorphic vector-valued modular forms for ϱ and let M(H) denote the collection of modular forms on H. Then M(ϱ) is a ℤ-graded M(H)-module. It has been proven that M(ϱ) may not be projective as a M(H)-module. We prove that M(ϱ) is Cohen-Macaulay as a M(H)-module. We also explain how to apply this result to prove that if M(H) is a polynomial ring, then M(ϱ) is a free M(H)-module of rank dim ϱ.

Similarity isometries of point packings.

A linear isometry R of {\bb R}^{d} is called a similarity isometry of a lattice \Gamma\subseteq{\bb R}^{d} if there exists a positive real number β such that βRΓ is a sublattice of (finite index in)Γ. The set βRΓ is referred to as a similar sublattice of Γ. A (crystallographic) point packing generated by a lattice Γ is a union of Γ with finitely many shifted copies ofΓ. In this study, the notion of similarity isometries is extended to point packings. A characterization for the similarity isometries of point packings is provided and the corresponding similar subpackings are identified. Planar examples are discussed, namely the 1 × 2 rectangular lattice and the hexagonal packing (or honeycomb lattice). Finally, similarity isometries of point packings about points different from the origin are considered by studying similarity isometries of shifted point packings. In particular, similarity isometries of a certain shifted hexagonal packing are computed and compared with those of the hexagonal packing.

Relationships between bounded languages, counter machines, finite-index grammars, ambiguity, and commutative regularity

It is shown that for every language family that is a trio containing only semilinear languages, all bounded languages in it can be accepted by one-way deterministic reversal-bounded multicounter machines (DCM). This implies that for every semilinear trio (where these properties are effective), it is possible to decide containment, equivalence, and disjointness concerning its bounded languages. A condition is also provided for when the bounded languages in a semilinear trio coincide exactly with those accepted by DCM machines, and it is used to show that many grammar systems of finite index — such as finite-index matrix grammars (Mfin) and finite-index ET0L (ET0Lfin) — have identical bounded languages as DCM.Then connections between ambiguity, counting regularity, and commutative regularity are made, as many machines and grammars that are unambiguous can only generate/accept counting regular or commutatively regular languages. Thus, such a system that can generate/accept a non-counting regular or non-commutatively regular language implies the existence of inherently ambiguous languages over that system. In addition, it is shown that every language generated by an unambiguous Mfin has a rational characteristic series in commutative variables, and is counting regular. This result plus the connections are used to demonstrate that the grammar systems Mfin and ET0Lfin can generate inherently ambiguous languages (over their grammars), as do several machine models. It is also shown that all bounded languages generated by these two grammar systems (those in any semilinear trio) can be generated unambiguously within the systems. Finally, conditions on Mfin and ET0Lfin languages implying commutative regularity are obtained. In particular, it is shown that every finite-index ED0L language is commutatively regular.

On Selmer Groups in the Supersingular Reduction Case

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all $\mathbb{Z}_p$-extensions of $F$ when they are defined.