Related papers: Classification of finite groups generated by refle…
In this paper, we survey some of the recent advances on embeddings into finitely generated (left-orderable) simple group such that the overgroup preserves algorithmic, geometric, or algebraic information about the embedded group. We discuss…
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…
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 $\mathcal{UV}$, where $\mathcal{U}$ (respectively,…
The orbits space of an irreducible representation of a finite group is a variety whose coordinate ring is finitely generated by homogeneous invariant polynomials. Boris Dubrovin showed that the orbits spaces of the reflection groups acquire…
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.
Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special…
We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem…
Several classes of irreducible orthogonal representations of compact Lie groups that are of importance in Differential Geometry have the property that the second osculating spaces of all of their nontrivial orbits coincide with the…
In this paper, we explore when a locally finite triangulated category has dimension zero or finite representation type. We also study generation of derived categories by orthogonal subcategories.
This paper investigates the finite generation of cluster automorphism groups. By applying the pseudo $\mathbb{N}$-grading introduced in our previous work, we establish a sufficient condition for a cluster automorphism group to be finitely…
We study and classify a class of representations (called generalized geometric representations) of a Coxeter group of finite rank. These representations can be viewed as a natural generalization of the geometric representation. The…
We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a…
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an…
We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of…
We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry…
We perform the computations necessary to establish a multiplicity one statement for the irreducible representations of a finite spin group which in turn yields the classification of irreducible representations of finite spin groups. (The…
We complete the description of group gradings on finite-dimensional incidence algebras. Moreover, we classify the finite-dimensional graded algebras that can be realized as incidence algebras endowed with a group grading.
We examine configurations of finite subsets of manifolds within the homotopy-theoretic context of $\infty$-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration…
For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the…
We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order…