Products and polarizations of super-isolated abelian varieties

In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.

Profinite genus of the fundamental groups of compact flat manifolds with holonomy group of prime order

In this article, we calculate the profinite genus of the fundamental group of an 𝑛-dimensional compact flat manifold 𝑋 with holonomy group of prime order. As consequence, we prove that if n ⩽ 21 n\leqslant 21 , then 𝑋 is determined among all 𝑛-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of the profinite completion of the fundamental group of 𝑋 in terms of the representation genus of its holonomy group.

Classifying gyrotransversals in groups

In this paper, we classify the gyrotransversals upto isomorphism in a group [Formula: see text] to a fixed subgroup of it. As an application, we have calculated the isomorphism class of gyrotransversals in finite dihedral group to any subgroup of it. As a consequence, we get a lower bound for non-isomorphic right gyrogroups of order [Formula: see text].

Condensed groups in product varieties

A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U ⁢ V \mathcal{UV} , where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U ⁢ V \mathcal{UV} .

Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams

The present paper gives a partial answer to Smale’s question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by “Smale surgery” are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}\subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.

Galois action on Fuchsian surface groups and their solenoids

Let C be a complex algebraic curve uniformized by a Fuchsian group Γ. In the first part of this paper we identify the automorphism group of the solenoid associated with Γ with the Belyaev completion of its commensurator {\mathrm{Comm}(\Gamma)} and we use this identification to show that the isomorphism class of this completion is an invariant of the natural Galois action of {\mathrm{Gal}(\mathbb{C}/\mathbb{Q})} on algebraic curves. In turn, this fact yields a proof of the Galois invariance of the arithmeticity of Γ independent of Kazhhdan’s. In the second part we focus on the case in which Γ is arithmetic. The list of further Galois invariants we find includes: (i) the periods of {\mathrm{Comm}(\Gamma)}, (ii) the solvability of the equations {X^{2}+\sin^{2}\frac{2\pi}{2k+1}} in the invariant quaternion algebra of Γ and (iii) the property of Γ being a congruence subgroup.

The Automorphism Group of Green Algebra of 9-Dimensional Taft Hopf Algebra

Let H3 be the 9-dimensional Taft Hopf algebra, let [Formula: see text] be the corresponding Green ring of H3, and let [Formula: see text] be the automorphism group of Green algebra [Formula: see text] over the real number field ℝ. We prove that the quotient group [Formula: see text] is isomorphic to the direct product of the dihedral group of order 12 and the cyclic group of order 2, where T1 is the isomorphism class which contains the identity map and is isomorphic to a group [Formula: see text] with multiplication given by [Formula: see text].

Reconstructing in the Constraint Satisfaction Problem

It is a long-standing problem in graph theory to prove or disprove the \emph{reconstruction conjecture}, also known as the Kelly-Ulam conjecture. This conjecture states that every simple graph on at least three vertices is \emph{reconstructible}, which means that the isomorphism class of such a graph is uniquely determined by the isomorphism classes of its vertex-deleted subgraphs. In this talk, the notion of reconstructing is extended from graphs to instances of the constraint satisfaction problem (CSP): an instance is \emph{reconstructible} if its isomorphism class is uniquely determined by the isomorphism classes of its constraint-deleted subinstances. Questions of interest include not only questions about reconstructible CSP instances but also about CSP instances with reconstructible properties and parameters such as the existence of solutions or the number of solutions. As shown in the talk, such questions can be answered using techniques borrowed and adapted from graph reconstruction. In particular, Lov\'{a}sz's method of counting graph homomorphisms \cite{Lov72} is adapted to characterize CSP instances for which the number of solutions is reconstructible.

Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

We compute the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.