Related papers: Division in group rings of surface groups

We investigate bounds on the dimension of cohomology groups for finite groups acting on an irreducible kG-module for G a finite group of bound sectional p-rank and k an algebraically closed field of characteristic p.

In this paper, we show the fundamental theorems for rotationally symmetric hypersurfaces, and thus, together with the earlier results in [3] and [4], provide a complete classification of umbilic hypersurfaces in the Heisenberg groups…

We describe the ring structure of the rational cohomology of the Torelli groups of the manifolds $\#^g S^n \times S^n$ in a stable range, for $2n \geq 6$. Some of our results are also valid for $2n=2$, where they are closely related to…

We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of algebraic surfaces. This gives a positive answer to a question of Vladimir L. Popov.

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed…

We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…

Given a finitely presented group $G$, Hopf's formula expresses the second integral homology of $G$ in terms of generators and relators. We give an algorithm that exploits Hopf's formula to estimate $H_2(G;k)$, with coefficients in a finite…

We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on…

We introduce a cohomology set for groups defined by algebraic difference equations and show that it classifies torsors under the group action. This allows us to compute all torsors for large classes of groups. We also develop some tools for…

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on…

We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves…

This paper proves an integral version of the Riemann-Roch theorem for surface bundles, comparing the standard cohomology classes with the cohomology classes coming from the symplectic group.

We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded…

A general method of computing cohomology groups of the space of nonsingular algebraic hypersurfaces of degree $d$ in $CP^n$ is described. Using this method, rational cohomology groups of such spaces with $n=2, d \le 4$ and $n=3=d$ are…

The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here "compute" means to find a presentation in terms of generators and relations, and involves only the…

We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates…

We characterize helix surfaces (constant angle surfaces) in the special linear group $\mathrm{SL}(2,\r)$. In particular, we give an explicit local description of these surfaces in terms of a suitable curve and a 1-parameter family of…

We discuss the applications of fundamental groups (of complements of curves) computations (and possibly the computations of the second homotopy group as a model over it) to the classification of algebraic surface. We prove that the…

This paper studies the class group of graded integral domains. As an application, we state a decomposition theorem for class groups of semigroup rings. This recovers well-known results developed for the classic contexts of polynomial rings…

Given an arrangement of subtori of arbitrary codimension in a torus, we compute the cohomology groups of the complement. Then, using the Leray spectral sequence, we describe the multiplicative structure on the graded cohomology. We also…