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We study a family of Zariski dense finitely generated discrete subgroups of $\mathrm{Isom}(\mathbb{H}^d)$, $d \geqslant 2$, defined by the following property: any group in this family contains at least one reflection in a hyperplane. As an…
We define and study sl\_2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for…
We study square lattice space group symmetry fractionalization in a family of exactly solvable models with $\mathbb{Z}_2$ topological order in two dimensions. In particular, we have obtained a complete understanding of which distinct types…
We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic…
In this second part we study first the group $Aut_{\mathbb Q}(S)$ of numerically trivial automorphisms of an algebraic properly elliptic surface $S$, that is, of a minimal algebraic surface with Kodaira dimension $\kappa(S)=1$, in the case…
We demonstrate the semiclassical nature of symmetry twist defects that differ from quantum deconfined anyons in a true topological phase by examining non-abelian crystalline defects in an abelian lattice model. An underlying non-dynamical…
We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…
We study the possibility of applying a finite-dimensionality argument in order to address parts of the Baum-Connes conjecture for finitely generated linear groups. This gives an alternative approach to the results of Guentner, Higson, and…
The cohomology of coherent sheaves and sheaves of Abelian groups on Noetherian schemes are interpreted in second order arithmetic by means of a finiteness theorem. This finiteness theorem provably fails for the etale topology even on…
In recent years, attempts to generalize lattice gauge theories to model topological order have been carried out through the so called $2$-gauge theories. These have opened the door to interesting new models and new topological phases which…
In this paper, we associate a family of infinite-rank pro-Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac--Moody algebra. In the case that the element has a polynomial…
In this paper, we determine the reduced automorphism groups of hyperelliptic curves of a small genus in characteristic $2$, when they are of $2$-rank $0$. Such a curve is an Artin-Schreier curve defined in the form $y^2-y=f(x)$ for a…
We determine the structure of automorphism groups of finite graphs of bounded Hadwiger number. Our proof includes a structural analysis of finite edge-transitive graphs. In particular, we show that for connected, $K_{h+1}$-minor-free,…
A smooth cuboid can be identified with a $3\times 3$ matrix of linear forms, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over…
We show that every Coxeter group that is not virtually abelian and for which all labels in the corresponding Coxeter graph are powers of 2 or infinity can be mapped onto uncountably many infinite 2-groups which, in addition, may be chosen…
A distributive lattice with zero is completely normal if its prime ideals form a root system under set inclusion.Every such lattice admits a binary operation (x,y) \mapsto x-y satisfying the rules x \leq y\vee (x-y) and (x-y) \wedge (y-x)=0…
It is well known that the discrete analogue of a lattice is a linear code which is a vector subspace of Hamming space $\mathbb{F}^n$. The set $\mathbb{F}$ is a finite field and $n \in \mathbb{Z}_{>0}$. Our attempt is to construct a class of…
We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by…
Let $(K,\mathcal O, k)$ be a $p$-modular system with $k$ algebraically closed and $\mathcal O$ unramified, and let $\Lambda$ be an $\mathcal O$-order in a separable $K$-algebra. We call a $\Lambda$-lattice $L$ rigid if ${\rm…
We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups…