Related papers: The relational complexity of linear groups acting …

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the…

To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors…

In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if $L$ is a nontrivial limit group then the…

Let $\text{AGL}(1,\Bbb F_q)$ denote the affine linear group of dimension one over the finite field $\Bbb F_q$. We determine the M\"obius function of the lattice of subgroups of $\text{AGL}(1,\Bbb F_q)$.

Let $p$ be a prime and $\mathbb{F}_p$ be a finite field of $p$ elements. Let $\mathbb{F}_pG$ denote the group algebra of the finite $p$-group $G$ over the field $\mathbb{F}_p$ and $V(\mathbb{F}_pG)$ denote the group of normalized units in…

This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable…

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by…

We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be…

Let $G$ be a permutation group on a set $\Omega$ of size $t$. We say that $\Lambda\subseteq\Omega$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda$. We define the…

We give a classification of ordered five points in $\mathbb P^3$ under the diagonal action of $GL_4$ over an algebraically closed field of characteristic $0$, by an explicit description of the diagonal action of $GL_4$ on the quintuple of…

An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which…

Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence…

We calculate the one-loop effective potential of a self-interacting scalar field on the spacetime of the form $\reals^2\times H^2/\Gamma$. The Selberg trace formula associated with a co-compact discrete group $\Gamma$ in $PSL(2,\reals )$…

Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence…

At mesoscopic scales, the quantum corrected field equations of gravity should arise from extremizing, $\Omega$, the number of microscopic configurations of pre-geometric variables consistent with a given geometry. This $\Omega$, in turn, is…

Let $G$ be a finite simple graph. The line graph $L(G)$ represents the adjacencies between edges of $G$. We define first the line simplicial complex $\Delta_L(G)$ of $G$ containing Gallai and anti-Gallai simplicial complexes…

In this paper we present a new class of complexity measures, induced by a new data structure for representing $k$-valued functions (operations), called minor decision diagram. The results are presented in terms of Multi-Valued Logic…

We consider space functions $s(n)$ of finitely presented groups $G =< A\mid R> .$ (These functions have a natural geometric analog.) To define $s(n)$ we start with a word $w$ over $A$ of length at most $n$ equal to 1 in $G$ and use…

We consider the action of the finite matrix group $SL(m,Z_n)$ on the ring $Z_n^m$. We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A.A. Kirillov for $m=2$ and $n$…

The Lie algbera of a compact semisimple Lie group G is determined by the degrees of the irreducible representations of G. However, two different groups can have the same representation degrees.