Infinite Dihedral Group Research Articles

Nuclear Dimension of Subhomogeneous Twisted Groupoid C*-algebras and Dynamic Asymptotic Dimension

Abstract We characterize subhomogeneity for twisted étale groupoid $\text{C}^{*}$-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid $\text{C}^{*}$-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_{\infty }$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes _{r} D_{\infty }$ has finite nuclear dimension and is classifiable by its Elliott invariant.

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

Schur rings over infinite dihedral group

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

Spectral dynamics for the infinite dihedral group and the lamplighter group

Spectral dynamics for the infinite dihedral group and the lamplighter group

Representations of free products of semisimple algebras via quivers

We use quiver representation theory and King's notion of stable representations to study the (finite dimensional) representations of free products of semi-simple associative K-algebras over an algebraically closed field K. We obtain numerical criteria giving necessary conditions for a module over such a free product to be simple and prove that these conditions are sufficient for modules in general position. We also prove that modules in general position are always semisimple, although non-semisimple modules always exist when the algebra is infinite dimensional. With essentially a single exception, all module categories are strictly wild; moreover these categories admit no bounds on the dimensions of simple modules. We use quiver moduli spaces to derive a closed formula for the number of parameters needed to describe all simple modules in a given dimension. As a special case, we apply our results to free products of finite groups, and recover the representation theory of the projective modular group PSL2(Z) (strictly wild) and of the infinite dihedral group (tame).

The infinite dihedral group and K3 surfaces with Picard number 2

The automorphism group of a K3 surface with Picard number two is either the infinite cyclic group or the infinite dihedral group, if it is infinite. In this paper, we determine some conditions for a K3 surface of Picard number two to have the infinite dihedral automorphism group.

The AH conjecture for Cantor minimal dihedral systems

The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani–Rokhlin partitions adapted to such systems.

On continuous orbit equivalence rigidity for virtually cyclic group actions

We prove that any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group by certain other virtually cyclic groups, e.g., the direct product of the integer group with any non-abelian finite simple group.

Flip signatures

AbstractA $D_{\infty }$ -topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group $D_{\infty }$ . It is defined by two zero-one square matrices A and J satisfying $AJ=JA^{\textsf {T}}$ and $J^2=I$ . A flip signature is obtained from symmetric bilinear forms with respect to J on the eventual kernel of A. We modify Williams’ decomposition theorem to prove the flip signature is a $D_{\infty }$ -conjugacy invariant. We introduce natural $D_{\infty }$ -actions on Ashley’s eight-by-eight and the full two-shift. The flip signatures show that Ashley’s eight-by-eight and the full two-shift equipped with the natural $D_{\infty }$ -actions are not $D_{\infty }$ -conjugate. We also discuss the notion of $D_{\infty }$ -shift equivalence and the Lind zeta function.

On product-one sequences over subsets of groups

Let G be a group and G_0 subseteq G be a subset. A sequence over G_0 means a finite sequence of terms from G_0, where the order of elements is disregarded and the repetition of elements is allowed. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. We study algebraic and arithmetic properties of monoids of product-one sequences over finite subsets of G and over the whole group G, with a special emphasis on the infinite dihedral group.

The homology of the groupoid of the self-similar infinite dihedral group

We compute the $K$-theory of the $C^*$-algebra associated to the self-similar infinite dihedral group, and the homology of its associated étale groupoid. We see that the rational homology differs from the $K$-theory, strongly contradicting a conjecture posted by Matui. Moreover, we compute the abelianization of the topological full group of the groupoid associated to the self-similar infinite dihedral group.

Self-similarity and spectral dynamics

This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.

Aut-invariant quasimorphisms on free products

Let G=A *B be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups.

A COMBINATORIAL IDENTITY AND THE FINITE DUAL OF INFINITE DIHEDRAL GROUP ALGEBRA

In this paper, we find a combinatorial identity which is related to the multi-dimensional integral γ m in the study of divisor functions (see (1)). As an application, we determine the finite dual of the group algebra of infinite dihedral group.

Counting finite orbits for the flip systems of shifts of finite type

<p style='text-indent:20px;'>For a discrete system <inline-formula><tex-math id="M1">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula>, the flip system <inline-formula><tex-math id="M2">\begin{document}$ (X,T,F) $\end{document}</tex-math></inline-formula> can be regarded as the action of infinite dihedral group <inline-formula><tex-math id="M3">\begin{document}$ D_\infty $\end{document}</tex-math></inline-formula> on the space <inline-formula><tex-math id="M4">\begin{document}$ X $\end{document}</tex-math></inline-formula>. Under this action, <inline-formula><tex-math id="M5">\begin{document}$ X $\end{document}</tex-math></inline-formula> is partitioned into a set of orbits. We are interested in counting the finite orbits in this partition via the prime orbit counting function. In this paper, we prove the asymptotic behaviour of this counting function for the flip systems of shifts of finite type. The proof relies mostly on combinatorial calculations instead of the usual approach via zeta function. Here, we are able to obtain more precise asymptotic result for this <inline-formula><tex-math id="M6">\begin{document}$ D_\infty $\end{document}</tex-math></inline-formula>-action on shifts of finite type as compared to other group actions on systems available in the literature.

Symbolic Calculation and Invertibility of Convolution Operators on the Infinite Dihedral Group

Symbolic Calculation and Invertibility of Convolution Operators on the Infinite Dihedral Group

Representation theory of the algebra generated by a pair of complex structures

The objective of this paper is to determine the finite-dimensional, indecomposable representations of the algebra that is generated by two complex structures over the real numbers. Since the generators satisfy relations that are similar to those of the infinite dihedral group, we give the algebra the name [Formula: see text].

Hermitian Metric and the Infinite Dihedral Group

Для набора $A=(A_1,A_2,…,A_n)$ элементов унитальной банаховой алгебры $\mathcal B$ рассматривается ассоциированный с ним многопараметрический пучок $A(z)=z_1 A_1 + z_2 A_2 + …+z_n A_n$. Проективным спектром $P(A)$ этого набора называется множество точек $z\in \mathbb C^n$, для которых элемент $A(z)$ необратим. С помощью фундаментальной формы $\Omega _A=-\omega _A^*\wedge \omega _A$, где $\omega _A(z) = A^{-1}(z) dA(z)$ - форма Маурера-Картана, Р. Дуглас и Р. Янг определили и исследовали естественную эрмитову метрику на резольвентном множестве $P^c(A)=\mathbb{C}^n\setminus P(A)$. В настоящей работе данная метрика изучается в случае бесконечной диэдральной группы $D_\infty = \langle a,t\mid a^2=t^2 =1\rangle $ относительно левого регулярного представления $\lambda $. Для неоднородного пучка $R(z) = I+z_1\lambda (a)+z_2\lambda (t)$ метрика на $P^c(R)$ вычислена в явном виде и показано, что пополнение множества $P^c(R)$ относительно этой метрики есть $\mathbb{C}^2\setminus \{(\pm 1,0), (0,\pm 1)\}$. Исключительные точки $(\pm 1,0)$, $(0,\pm 1)$ соответствуют классическому спектру $\sigma (\lambda (a))=\sigma (\lambda (t))=\{\pm 1\}$. Настоящая работа является продолжением исследований Р. Дугласа, Р. Янга (2018) и Р.И. Григорчука, Р. Янга (2017).

Blob algebra and two-color Soergel calculus

In 2003, Martin and Woodcock noticed a connection between the representation theory of the blob algebra and the Kazhdan–Lusztig polynomials associated with the infinite dihedral group. However, no conceptual explanation for this coincidence has yet been provided. In this study, a possible explanation of this phenomenon is suggested by enunciating a conjecture that relates the endomorphism algebra of Bott–Samelson bimodules to certain subalgebras of the blob algebra obtained by idempotent truncation. Evidence supporting this conjecture is provided.

Hermitian Metric and the Infinite Dihedral Group

For a tuple A = (A1,A2,…, An) of elements in a unital Banach algebra B, the associated multiparameter pencil is A(z) = z1A1 + z2A2 + … + znAn. The projective spectrum P(A) is the collection of z ∈ ℂn such that A(z) is not invertible. Using the fundamental form ΩA = −ω * ∧ ωA, where ωA(z) = A−1(z) dA(z) is the Maurer–Cartan form, R. Douglas and the second author defined and studied a natural Hermitian metric on the resolvent set Pc(A) = ℂn \ P(A). This paper examines that metric in the case of the infinite dihedral group, D∞ = , with respect to the left regular representation λ. For the non-homogeneous pencil R(z) = I + z1λ(a) + z2λ(t), we explicitly compute the metric on Pc(R) and show that the completion of Pc(R) under the metric is ℂ2 \ {(±1, 0), (0, ±1)}, which rediscovers the classical spectra σ(λ(a)) = σ(λ(t)) = {± 1}. This paper is a follow-up of the papers by R. G. Douglas and R. Yang (2018) and R. Grigorchuk and R. Yang (2017).