Infinite Group Research Articles

On additive bases in infinite abelian semigroups

Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups and semigroups. We show that, for every infinite abelian group T , the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality k contained in an additive basis of order at most h can be bounded in terms of h and k alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon \mathbf{N} . Also, using invariant means, we address a classical problem, initiated by Erdős and Graham and then generalized by Nash and Nathanson both in the case of \mathbf{N} , of estimating the maximal order X_T(h,k) that a basis of cocardinality k contained in an additive basis of order at most h can have.Among other results, we prove that X_T(h,k)=O(h^{2k+1}) for every integer k \ge 1 . This result is new even in the case where k=1 . Besides the maximal order X_T(h,k) , the typical order S_T(h,k) is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in \mathbf{N} and in abelian groups.

Embedded surfaces with infinite cyclic knot group

Embedded surfaces with infinite cyclic knot group

Handle decompositions for a class of closed orientable PL 4-manifolds

In this article, we study a class of closed connected orientable PL $4$-manifolds admitting a semi-simple crystallization and which have an infinite cyclic fundamental group. We show that the manifold in the class admits a handle decomposition in which the number of $2$-handles depends upon its second Betti number and other $h$-handles ($h \leq 4$) are at most $2$. More precisely, our main result is the following. For a closed connected orientable PL $4$-manifold having a semi-simple crystallization with the fundamental group as $\mathbb{{Z}}$, we have constructed a handle decomposition for $M$ as one of the following types: $(1)$ one $0$-handle, two $1$-handles, $1+\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, $(2)$ one $0$-handle, one $1$-handle, $\beta_2(M)$ $2$-handles, one $3$-handle and one $4$-handle, where $\beta_2(M)$ denotes the second Betti number of manifold $M$ with $\mathbb{Z}$ coefficients.

On infinite group velocity of Lamb waves

The analytical and geometrical conditions for the possible appearance of the infinite group velocity (IGV) points belonging to the dispersion curves of Lamb waves propagating in traction-free plates are studied by a combined method comprising Cauchy sextic formalism and the exponential fundamental matrix method. According to the obtained geometrical condition, the IGV corresponds to the coincidence of a tangent line to any of the dispersion curves with a straight line passing through the origin. The developed technique is demonstrated by applying it to the analysis of Lamb wave dispersion in a Ge cubic crystal.

Spectral Methods from Tensor Networks

A tensor network is a diagram that specifies a way to “multiply” a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.

Ranks of 𝑅𝑂(𝐺)-graded stable homotopy groups of spheres for finite groups 𝐺

We describe the distribution of infinite groups within the R O ( G ) RO(G) -graded stable homotopy groups of spheres for a finite group G G .

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

Algebraic Morphology of DNA–RNA Transcription and Regulation

Transcription factors (TFs) and microRNAs (miRNAs) are co-actors in genome-scale decoding and regulatory networks, often targeting common genes. To discover the symmetries and invariants of the transcription and regulation at the scale of the genome, in this paper, we introduce tools of infinite group theory and of algebraic geometry to describe both TFs and miRNAs. In TFs, the generator of the group is a DNA-binding domain while, in miRNAs, the generator is the seed of the sequence. For such a generated (infinite) group π, we compute the SL(2,C) character variety, where SL(2,C) is simultaneously a ‘space-time’ (a Lorentz group) and a ‘quantum’ (a spin) group. A noteworthy result of our approach is to recognize that optimal regulation occurs when π looks similar to a free group Fr (r=1 to 3) in the cardinality sequence of its subgroups, a result obtained in our previous papers. A non-free group structure features a potential disease. A second noteworthy result is about the structure of the Groebner basis G of the variety. A surface with simple singularities (such as the well known Cayley cubic) within G is a signature of a potential disease even when π looks similar to a free group Fr in its structure of subgroups. Our methods apply to groups with a generating sequence made of two to four distinct DNA/RNA bases in {A,T/U,G,C}. We produce a few tables of human TFs and miRNAs showing that a disease may occur when either π is away from a free group or G contains surfaces with isolated singularities.

Relative uniformly positive entropy of induced amenable group actions

AbstractLet G be a countably infinite discrete amenable group. It should be noted that a G-system $(X,G)$ naturally induces a G-system $(\mathcal {M}(X),G)$ , where $\mathcal {M}(X)$ denotes the space of Borel probability measures on the compact metric space X endowed with the weak*-topology. A factor map $\pi : (X,G)\to (Y,G)$ between two G-systems induces a factor map $\widetilde {\pi }:(\mathcal {M}(X),G)\to (\mathcal {M}(Y),G)$ . It turns out that $\widetilde {\pi }$ is open if and only if $\pi $ is open. When Y is fully supported, it is shown that $\pi $ has relative uniformly positive entropy if and only if $\widetilde {\pi }$ has relative uniformly positive entropy.

On Numerical Dimensions of Calabi–Yau Varieties

Abstract Let $X$ be a Calabi–Yau variety of Picard number two with infinite birational automorphism group. We show that the numerical dimension $\kappa ^{\mathbb{R}}_{\sigma }$ of the extremal rays of the closed movable cone of $X$ is $\dim X/2$. More generally, we investigate the relation between the two numerical dimensions $\kappa ^{\mathbb{R}}_{\sigma }$ and $\kappa ^{\mathbb{R}}_{\textrm{vol}}$ for Calabi–Yau varieties. We also compute $\kappa ^{\mathbb{R}}_{\sigma }$ for non-big divisors in the closed movable cone of a projective hyperkähler manifold.

Unbounded domains in hierarchically hyperbolic groups

We investigate unbounded domains in hierarchically hyperbolic groups and obtain constraints on the possible hierarchical structures. Using these insights, we characterise the structures of virtually abelian HHGs and show that the class of HHGs is not closed under finite extensions. This provides a strong negative answer to the question of whether being an HHG is invariant under quasi-isometries. Along the way, we show that infinite torsion groups are not HHGs. By ruling out pathological behaviours, we are able to give simpler, direct proofs of the rank-rigidity and omnibus subgroup theorems for HHGs. This involves extending our techniques so that they apply to all subgroups of HHGs.

Geometric Structures in Group Theory

The conference was in the area of geometric group theory, the field of mathematics in which one studies infinite groups (finitely generated, or more generally locally compact, countable etc.) via actions on spaces endowed with various structures (geometric, measurable, analytic etc.). The surging current activity in the field is drawing more and more connections with other mathematical areas, and this was successfully reflected in the program of this week, during which problems in algebraic topology, representation theory and functional analysis, to name just a few, featured prominently alongside core topics in the area.

How to overcome the instability of AHP weight? —Partial order analysis method based on the lower limit of the proportion

In the AHP method, it is essential to control the uncertainty in obtaining weight. The paper aims to reduce the uncertainty introduced when using the AHP method, so as to improve the stability of the weight. Firstly, the paper ranks the indicators according to their importance, and the lowest lower limits of the importance ratio of two adjacent indicators are obtained, which can replace the accurate values of the indicator comparison in the AHP method and significantly reduce the introduced uncertainty. Secondly, the schemes are compared under infinite groups of weights, avoiding the blindness and instability of the comparison results obtained by only one group of weights. At the same time, the schemes’ comprehensive comparison results are displayed in the partial order Hasse diagram, which can clearly identify whether the comparison between scheme pairs is stable and evaluate the stability of the weight. Finally, an example is given to show the specific operation process of the proposed method, and simulation is used to verify the method’s superiority.

On infinite anticommutative groups

On infinite anticommutative groups

Symmetric skew braces and brace systems

Abstract For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ⁢ ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a map from ( G , ⋅ ) {(G,\cdot\,)} to Aut ⁡ ( G , ⋅ ) {\operatorname{Aut}(G,\cdot\,)} , which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : ( G , ⋅ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\cdot\,)\to\operatorname{Aut}(G,\cdot\,)} is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitutes a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite groups.

Concentration of measure for classical Lie groups

We study the concentration of measure in metric-measurable (mm)-spaces. We define the notion of concentration locus of a flag sequence of metric-measurable (mm)-spaces. Some examples of infinite group action on an infinite dimensional compact and non-compact manifold show the role played by the trajectory of concentration locus. We also provide some applications in physics, which emphasize the role of concentration of measure in gravitational effects.

Infinite density matrix renormalization group method for solving the Ising model

Infinite density matrix renormalization group method for solving the Ising model

On continuous orbit equivalence rigidity for virtually cyclic group actions

We prove that any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group by certain other virtually cyclic groups, e.g., the direct product of the integer group with any non-abelian finite simple group.

Probabilistically nilpotent groups of class two

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Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in Lie groups

For a locally compact Hausdorff group G and the compact space \mathrm{Sub}_G of closed subgroups of G endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of G on \mathrm{Sub}_G in terms of distality and expansivity. We prove that an infinite discrete group G , which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on \mathrm{Sub}^c_G , the space of cyclic subgroups of G , while only the finite order automorphisms of G act distally on \mathrm{Sub}^c_G . For an automorphism T of a connected Lie group G which keeps a lattice \Gamma invariant, we compare the behaviour of the actions of T on \mathrm{Sub}_G and \mathrm{Sub}_\Gamma in terms of distality. Under certain necessary conditions on the Lie group G , we show that T acts distally on \mathrm{Sub}_G if and only if it acts distally on \mathrm{Sub}_\Gamma . We also obtain certain results about the structure of lattices in a connected Lie group.