Finite General Groups Research Articles

Weak Shadowing for Actions of Some Finitely Generated Groups on Non-compact Spaces and Related Measures

In this paper, we introduce the notions of topological stability, shadowing, and weak shadowing properties for actions of some finitely generated groups on non-compact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Also, we extend Walter’s stability theorem to group actions on locally compact metric spaces. We show that action $\varphi :\mathbb {F}_{2}\times M\to M$ of free group F2 = on compact manifold M with dim(M) ≥ 2 has shadowing property if and only if it has weak shadowing property. This implies that if $\varphi :\mathbb {F}_{2}\times M\to M$ is topologically stable, then it has shadowing property. Finally, we introduce a measure $\mu \in {\mathscr{M}}(X)$ that is compatible with the weak shadowing property for the continuous action φ : G × X → X, denoted by $\mu \in {\mathscr{M}}(X, \varphi )$. We show that if φ can be approximated by μ-transitive actions for some $\mu \in {\mathscr{M}}(X, \varphi )$, then Ω(φ) = X. We give an example to show that it may be ${\mathscr{M}}(X, \varphi )\neq \emptyset $ while φ does not have weak shadowing property but we show that if there is a strictly measure $\mu \in {\mathscr{M}}(X, \varphi )$, then continuous action φ of G on compact metric space X has weak shadowing property. Also, we show that if G is a finitely generated abelian group and ${\mathscr{M}}(X, \varphi )\neq \emptyset $, then closure of ${\mathscr{M}}(X, \varphi )$ has an φ-invariant measure.

Positive speed self-avoiding walks on graphs with more than one end

A self-avoiding walk (SAW) is a path on a graph that visits each vertex at most once. The mean square displacement of an n-step SAW is the expected value of the square of the distance between its ending point and starting point, where the expectation is taken with respect to the uniform measure on n-step SAWs starting from a fixed vertex. It is conjectured that the mean square displacement of an n-step SAW is asymptotically n2ν, where ν is a constant. Computing the exact values of the exponent ν on various graphs has been a challenging problem in mathematical and scientific research for long.In this paper we show that on any locally finite Cayley graph of an infinite, finitely-generated group with more than two ends, the number of SAWs whose end-to-end distances are linear in lengths has the same exponential growth rate as the number of all the SAWs. We also prove that for any infinite, finitely-generated group with more than one end, there exists a locally finite Cayley graph on which SAWs have positive speed - this implies that the mean square displacement exponent ν=1 on such graphs.These results are obtained by proving more general theorems for SAWs on quasi-transitive graphs with more than one end, which make use of a variation of Kesten's pattern theorem in a surprising way, as well as the Stalling's splitting theorem. Applications include proving that SAWs have positive speed on the square grid in an infinite cylinder, and on the infinite free product graph of two connected, quasi-transitive graphs.

Collectives of Automata in Finitely Generated Groups

Данная работа посвящена обхождению лабиринта коллективом конечных автоматов. Эта часть теории автоматов породила довольно широкий спектр различных задач [1], [2], в том числе связанных с задачами теории сложности вычислений и теорией вероятностей. Оказывается, что рассмотрение сложных алгебраических объектов, таких как бернсайдовы группы, может быть интересным в данном контексте. В работе будет показано, что граф Кэли конечно-порожденной группы нельзя обойти коллективом автоматов тогда и только тогда, когда она бесконечна и каждый ее элемент периодичен. Библиография: 18 названий.

Group stability and Property (T)

In recent years, there has been a considerable amount of interest in the stability of a finitely-generated group Γ with respect to a sequence of groups {Gn}n=1∞, equipped with bi-invariant metrics {dn}n=1∞. We consider the case Gn=U(n) (resp. Gn=Sym(n)), equipped with the normalized Hilbert-Schmidt metric dnHS (resp. the normalized Hamming metric dnHamming). Our main result is that if Γ is infinite, hyperlinear (resp. sofic) and has Property (T), then it is not stable with respect to (U(n),dnHS) (resp. (Sym(n),dnHamming)).This answers a question of Hadwin and Shulman regarding the stability of SL3(Z). We also deduce that the mapping class group MCG(g), g≥3, and Aut(Fn), n≥3, are not stable with respect to (Sym(n),dnHamming).Our main result exhibits a difference between stability with respect to the normalized Hilbert-Schmidt metric on U(n) and the (unnormalized) p-Schatten metrics, since many groups with Property (T) are stable with respect to the latter metrics, as shown by De Chiffre-Glebsky-Lubotzky-Thom and Lubotzky-Oppenheim.We suggest a more flexible notion of stability that may repair this deficiency of stability with respect to (U(n),dnHS) and (Sym(n),dnHamming).

Multiplicative Dependence Among Iterated Values of Rational Functions Modulo Finitely Generated Groups

AbstractWe study multiplicative dependence between elements in orbits of algebraic dynamical systems over number fields modulo a finitely generated multiplicative subgroup of the field. We obtain a series of results, many of which may be viewed as a blend of Northcott’s theorem on boundedness of preperiodic points and Siegel’s theorem on finiteness of solutions to $S$-unit equations.

Counting inversions and descents of random elements in finite Coxeter groups

We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.

Groups of line and circle homeomorphisms. Criteria for almost nilpotency

For finitely-generated groups of line and circle homeomorphisms a criterion for their being almost nilpotent is established in terms of free two-generator subsemigroups and the condition of maximality. Previously the author found a criterion for almost nilpotency stated in terms of free two-generator subsemigroups for finitely generated groups of line and circle homeomorphisms that are -smooth and mutually transversal. In addition, for groups of diffeomorphisms, structure theorems were established and a number of characteristics of such groups were proved to be typical. It was also shown that, in the space of finitely generated groups of -diffeomorphisms with a prescribed number of generators, the set of groups with mutually transversal elements contains a countable intersection of open dense subsets (is residual). Navas has also obtained a criterion for the almost nilpotency of groups of -diffeomorphisms of an interval, where , in terms of free subsemigroups on two generators. Bibliography: 21 titles.

The Threshold for Subgroup Profiles to Agree is Logarithmic

For primes $p,e>2$ there are at least $p^{e-3}/e$ groups of order $p^{2e+2}$ that have equal multisets of isomorphism types of proper subgroups and proper quotient groups, isomorphic character tables, and power maps. This obstructs recent speculation concerning a path towards efficient isomorphism tests for general finite groups. These groups have a special purpose polylogarithmic-time isomorphism test.

On periodic groups of homeomorphisms of the 2–dimensional sphere

We prove that every finitely-generated group of homeomorphisms of the 2-dimensional sphere all of whose elements have a finite order which is a power of 2 and so that there exists a uniform bound for the order of group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of 2 provided there is an element of even order.

On Torsion and Finite Extension of $FC$ and $\tau N_{K}$ Groups in Certain Classes of Finitely Generated Groups

On Torsion and Finite Extension of $FC$ and $\tau N_{K}$ Groups in Certain Classes of Finitely Generated Groups

Finite Groups without Elements of Order Six

In 1977, in three papers by Podufalov, by Gordon, and by Fletcher, Stellmacher, and Stewart, finite simple groups without elements of order 6 were determined independently. In the present paper, using this result, we obtain a sufficiently complete description of the structure of a general finite group with this property.

The Smallest Singular Values and Vector-Valued Jack Polynomials

There is a space of vector-valued nonsymmetric Jack polynomials associated with any irreducible representation of a symmetric group. Singular polynomials for the smallest singular values are constructed in terms of the Jack polynomials. The smallest singular values bound the region of positivity of the bilinear symmetric form for which the Jack polynomials are mutually orthogonal. As background there are some results about general finite reflection groups and singular values in the context of standard modules of the rational Cherednik algebra.

Symmetric complete sum-free sets in cyclic groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0, 1/3], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of ℤp of size at least (1/3−c)·p, where c > 0 is a universal constant.

The Absolute of Finitely Generated Groups: II. The Laplacian and Degenerate Parts

The article continues the series of papers on the absolute of finitely generated groups. The absolute of a group with a fixed system of generators is defined as the set of ergodic Markov measures for which the system of cotransition probabilities is the same as for the simple (right) random walk generated by the uniform distribution on the generators. The absolute is a new boundary of a group, generated by random walks on the group. We divide the absolute into the Laplacian part and degenerate part and describe the connection between the absolute, homogeneous Markov processes, and the Laplace operator; prove that the Laplacian part is preserved under tak- ing some central extensions of groups; reduce the computation of the Laplacian part of the absolute of a nilpotent group to that of its abelinization; consider a number of fundamental examples (free groups, commutative groups, the discrete Heisenberg group). Keywords: absolute, Laplace operator, dynamic Cayley graph, nilpotent groups, Laplacian part of absolute.

On a Banach Algebra Generated by the Bergman Operator, Constant Coefficients, and Finitely Generated Groups of Shifts

We study a Banach algebra generated by the Bergman operator, constant coefficients, and shifts formed by finitely generated commutative groups of hyperbolic transformations of a unit disk acting in the Lebesgue space L p , p > 1, and obtain an efficient criterion for the operators from the considered Banach algebra to be Fredholm operators.

James' Submodule Theorem and the Steinberg Module

James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split $BN$-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor.

Boundary Convex Cocompactness and Stability of Subgroups of Finitely Generated Groups

Abstract A Kleinian group $\Gamma < \mathrm{Isom}(\mathbb H^3)$ is called convex cocompact if any orbit of $\Gamma$ in $\mathbb H^3$ is quasiconvex or, equivalently, $\Gamma$ acts cocompactly on the convex hull of its limit set in $\partial \mathbb H^3$. Subgroup stability is a strong quasiconvexity condition in finitely generated groups which is intrinsic to the geometry of the ambient group and generalizes the classical quasiconvexity condition above. Importantly, it coincides with quasiconvexity in hyperbolic groups and convex cocompactness in mapping class groups. Using the Morse boundary, we develop an equivalent characterization of subgroup stability which generalizes the above boundary characterization from Kleinian groups.

Symmetric Complete Sum-free Sets in Cyclic Groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0,13], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of Zp of size at least (13−c)⋅p, where c>0 is a universal constant.

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group [Formula: see text] into conjugate subgroups, [Formula: see text] for [Formula: see text] and [Formula: see text], where [Formula: see text] is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of [Formula: see text]. We also show that every solvable group [Formula: see text] is a product of at most [Formula: see text] conjugates of a Carter subgroup [Formula: see text] of [Formula: see text], where [Formula: see text] is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.

The spectrum of the equivariant stable homotopy category of a finite group

We study the spectrum of prime ideals in the tensor-triangulated category of compact equivariant spectra over a finite group. We completely describe this spectrum as a set for all finite groups. We also make significant progress in determining its topology and obtain a complete answer for groups of square-free order. For general finite groups, we describe the topology up to an unresolved indeterminacy, which we reduce to the case of p-groups. We then translate the remaining unresolved question into a new chromatic blue-shift phenomenon for Tate cohomology. Finally, we draw conclusions on the classification of thick tensor ideals.