How to generalize known results on equations over groups

Notes on Actions of Sets and Groups and Generalized Affine Spaces

Some well-known and less well-known or new notions related to group actions are surveyed. Some of these notions are used to generalize affine spaces. Actions are seen as functions with values in transformation monoids

Hyperbolic generalized triangle groups, property (T) and finite simple quotients

We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdan's property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Kazhdan groups possess finite simple quotient groups of arbitrarily large rank; they constitute the first known specimens combining those properties. All the hyperbolic groups we consider are non-positively curved k-fold generalized triangle groups, i.e. groups that possess a simplicial action on a CAT(0) triangle complex, which is sharply transitiv…

Finite groups in which generalized normality is a transitive relation

In this paper, we discuss some well-known results and some open problems of the theory of $σ$-properties of a group related to the study of generalized $T$-groups.

A new algorithm for fast generalized DFTs

We give an new arithmetic algorithm to compute the generalized Discrete Fourier Transform (DFT) over finite groups $G$. The new algorithm uses $O(|G|^{ω/2 + o(1)})$ operations to compute the generalized DFT over finite groups of Lie type, including the linear, orthogonal, and symplectic families and their variants, as well as all finite simple groups of Lie type. Here $ω$ is the exponent of matrix multiplication, so the exponent $ω/2$ is optimal if $ω= 2$. Previously, "exponent one" algorithms were known for supersolvable groups and the symmetric and alternating groups. No exponent one algorit…

Generalized torsion elements in the knot groups of twist knots

It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot $5_2$, which is the $(-2)$-twist knot, by Naylor and Rolfsen.

sofic群的广义Gottschalk猜想及其应用

Generalized Gottschalk's conjecture for sofic groups and applications

该研究将元胞自动机的满射性与对偶满射性定理推广到sofic群上元胞自动机的局部及全局一致有界奇异扰动情形，得到代数非均匀元胞自动机的满射性，并证明sofic群上扭群环的稳定有限性，推广了Kaplansky相关猜想的已知结果。