Dense Subgroup Research Articles

Topology and Automorphism Structures in General Linear Group

The general linear group, as a significant topic in algebra and topology, has a wide-ranging research background and application value. With the continuous advancement of mathematical research, the combination of topological structures and automorphism structures has become a key entry point for exploring the properties of the general linear group. This paper investigates the topological properties of general linear group GL (n, R) , specifically focusing on compactness, connectedness and the fundamental group, and the automorphism. The first part of the study is dedicated to a detailed analysis of these properties, providing new insights into the topological structural characteristics of GL (n, R) . This paper scrutinized the homotopy type within it, giving proof to the fundamental group of GL (n, R) , which is showed to be isomorphism to a trivial group. In the second half, This paper introduce and examine the function φ (G A GL n G A G ) = { ( , )| • } ∈ =  , which is used to identify the automorphism group of a dense subgroup G of Rn . Specifically, the automorphism group of Qn is investigated. This paper show that the automorphism group of it is isomorphic to the subgroup GL (n, Q) of general linear group GL (n, R) . Through this function, this paper offer an innovative approach to understanding the automorphisms within general linear groups, revealing the deep connections between algebraic and topological aspects of these groups.

On invariant means and pre-syndetic subgroups

Beyond the locally compact case, equivalent notions of amenability diverge, and some properties no longer hold, for instance amenability is not inherited by topological subgroups. This investigation is guided by some amenability-type properties of groups of paths and loops. It is shown that a version of amenability called skew-amenability is inherited by pre-syndetic subgroups in the sense of Basso and Zucker (in particular, by co-compact subgroups). It follows that co-compact subgroups of amenable topological groups whose left and right uniformities coincide are amenable. We discuss a version of amenability belonging to P. Malliavin and M.-P. Malliavin: the existence of a mean on bounded Borel functions that is invariant under the left action of a dense subgroup. We observe that this property is in general strictly stronger than amenability, and establish for it Reiter- and Følner-type criteria. Finally, there is a review of open problems.

Countable dense homogeneity property of connected compact manifolds

Let X be a connected compact manifold and H(X) denote the set of all homeomorphisms from X onto itself. In this paper we provide another proof for the known result that every connected compact manifolds are n−homogeneous and thereby countable dense homogeneous (CDH). We also show that every compact manifold can be expressed as a disjoint union of countable dense subsets. Additionally, we prove that for every h ∈ H(X), ∃ a partition of X in to countable dense subsets on which it is invariant. Mainly, we discuss about the denseness of HA(X) = { h ∈ H(X) | h : A → A is a homeomorphism } in H(X) whenever A is a subset of X. Finally, we prove that if f is a contraction function on X, then the set of all homeomorphisms that commutes with f is a nowhere dense subgroup of H(X).

Non-concentration property of Patterson–Sullivan measures for Anosov subgroups

Abstract Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $ , we prove that any $\Gamma $ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ . More generally, we prove that for a Zariski dense $\theta $ -transverse subgroup $\Gamma <G$ , any $(\Gamma , \psi )$ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ , provided the $\psi $ -Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

Zariski dense surface subgroups in with odd

AbstractFor odd n we construct a path $\rho\;:\;\thinspace \Pi_1(S) \to SL(n\mathbb{R})$ of discrete, faithful, and Zariski dense representations of a surface group such that $\rho_t(\Pi_1(S)) \subset SL(n,\mathbb{Q})$ for every $t\in \mathbb{Q}$ .

BASIC NONARCHIMEDEAN JØRGENSEN THEORY

Abstract We prove a nonarchimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of ${\mathrm {SL}_2}(K)$ , where K is a p-adic field, contains two elements that generate a dense subgroup of ${\mathrm {SL}_2}(K)$ , which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, J. Algebra261(2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of ${\mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete.

Effective Density of Non-Degenerate Random Walks on Homogeneous Spaces

Abstract We prove effective density of random walks on homogeneous spaces, assuming that the underlying measure is supported on matrices generating a dense subgroup and having algebraic entries. The main novelty is an argument passing from high dimension to effective equidistribution in the setting of random walks on homogeneous spaces, exploiting the spectral gap of the associated convolution operator.

On denseness of horospheres in higher rank homogeneous spaces

Abstract Let $ G $ be a connected semisimple real algebraic group and $\Gamma <G$ be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of N. Let $\mathcal E$ denote the unique P-minimal subset of $\Gamma \backslash G$ and let $\mathcal E_0$ be a $P^\circ $ -minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal E_0$ : (1) $gP\in G/P$ is a horospherical limit point; (2) $[g]NM$ is dense in $\mathcal E$ ; (3) $[g]N$ is dense in $\mathcal E_0$ . The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the $NM$ -minimality of $\mathcal E$ does not hold in a general Anosov homogeneous space.

The Frattini subgroup of a Lie group and the topological rank of a Lie group

We define the Frattini subgroup of a topological group G as the set of non-generators. We describe it completely for connected Lie groups, first for the case that G is solvable and then in general. As an application we determine the topological rank of G, that is the minimal number of elements needed to generate a dense subgroup of G.

Dichotomy and Measures on Limit Sets of Anosov Groups

Abstract Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$, we show that a $\Gamma $-conformal measure is supported on the limit set of $\Gamma $ if and only if its dimension is $\Gamma $-critical. This implies the uniqueness of a $\Gamma $-conformal measure for each critical dimension, answering the question posed in our earlier paper with Edwards [13]. We obtain this by proving a higher rank analogue of the Hopf–Tsuji–Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups.

Quasi-invariant measures on topological groups and ω-powers

Abstract Under GCH, there are described the cardinalities of all Hausdorff topological groups G such that there is a nonzero Borel measure on G having the card ⁢ ( G ) {{\rm card}(G)} -Suslin property and quasi-invariant with respect to an everywhere dense subgroup of G. Some connections are pointed out with the method of Kodaira and Kakutani (1950) for constructing a nonseparable translation invariant extension of the standard Lebesgue (Haar) measure on the circle group 𝐒 1 {{\bf S}_{1}} .

Strongly dense free subgroups of semisimple algebraic groups II

It was shown in [9] that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least 1 in characteristic 0 and at least 2 in positive characteristic. We also consider related questions for surface groups.

Simplifying social networks via triangle-based cohesive subgraphs

One main challenge for simplifying node-link diagrams of large-scale social networks lies in that simplified graphs generally contain dense subgroups or cohesive subgraphs. Graph triangles quantify the solid and stable relationships that maintain cohesive subgraphs. Understanding the mechanism of triangles within cohesive subgraphs contributes to illuminating patterns of connections within social networks. However, prior works can hardly handle and visualize triangles in cohesive subgraphs. In this paper, we propose a triangle-based graph simplification approach that can filter and visualize cohesive subgraphs by leveraging a triangle-connectivity called k-truss and a force-directed algorithm. We design and implement TriGraph, a web-based visual interface that provides detailed information for exploring and analyzing social networks. Quantitative comparisons with existing methods, two case studies on real-world datasets, and feedback from domain experts demonstrate the effectiveness of TriGraph.

Boundaries of dense subgroups of totally disconnected groups

Let Γ \Gamma be a countable discrete group and let H H be a lcsc totally disconnected group, L L a compact open subgroup of H H , and ρ : Γ → H \rho : \Gamma \rightarrow H a homomorphism with dense image. In this paper we construct, for every bi- L L -invariant probability measure θ \theta on H H , an explicit Furstenberg discretization τ \tau of θ \theta such that the Poisson boundary ( B θ , ν θ ) (B_\theta ,\nu _\theta ) of ( H , θ ) (H,\theta ) is a τ \tau -boundary, where Γ \Gamma acts on B θ B_\theta via the homomorphism ρ \rho . We also provide several criteria for when this τ \tau -boundary is maximal. Our technique can for instance be used to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not L p L^p -irreducible for any p ≥ 1 p \geq 1 , answering a conjecture of Bader-Muchnik in the negative. Furthermore, we provide the first example of a countable discrete group Γ \Gamma and two spread-out probability measures τ 1 \tau _1 and τ 2 \tau _2 on Γ \Gamma such that the boundary entropy spectrum of ( Γ , τ 1 ) (\Gamma ,\tau _1) is an interval, while the boundary entropy spectrum of ( Γ , τ 2 ) (\Gamma ,\tau _2) is a Cantor set.

Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties are preserved.

Coiling embolization strategy for medium-to-giant-sized intracranial aneurysms treated with pipeline embolization device: a propensity score-weighted study.

We aim to investigate associations between different coil strategies and outcomes in the aneurysms treated by a pipeline embolization device (PED). Patients with medium-to-giant-sized aneurysms treated by PED were included. The total cohort was divided into PED-alone and PED-coiling groups, and the PED-coiling group was further divided into loose and dense packing subgroups. Multivariate logistic analyses and stabilized inverse probability of treatment weighting (sIPTW) were performed to investigate the relationships between coiling strategies and outcomes. Restricted cubic spline (RCS) curves were used to describe the coiling degree and angiographic outcome relationship. A total of 398 patients with 410 aneurysms were included. Aneurysms treated with PED coiling had a lower incomplete occlusion rate (15.3% vs. 30.3%, p = 0.002), higher total perioperative complication rate (14.2% vs. 3.5%, p = 0.001), longer production time (142.14min vs. 101.26min, p < 0.001), and higher total cost ($45,158.63 vs. $34,680.91, p < 0.001) than those treated with PED alone. There were no differences in outcomes between the loose and dense packing subgroups. However, the total cost was higher in the dense packing group ($43,787.46 vs. $47,288.32, p = 0.001) than in the loose packing group. The result was still robust in the multivariate and sIPTW analyses. The RCS curves showed "L-shape" relationships between the coil degree and angiographic outcomes. Compared with PED alone, PED coiling could improve aneurysm occlusion. However, it could also increase the total complication risk, prolong procedure time, and increase the total cost. Compared with loose packing, dense packing did not enhance the treatment effectiveness but increased the treatment cost. The additional treatment effect from coiling embolization declines sharply after a certain point. Specifically, the aneurysm occlusion rate is roughly stable when the coil number is greater than 3 or the total coil length is longer than 150cm. • Compared with pipeline embolization device (PED) alone, PED combined with coiling can improve aneurysm occlusion. • Compared with PED alone, PED combined with coiling increases the total complication risk, cost, and prolongs procedure time. • Compared with loose packing, dense packing did not increase the treatment effectiveness but increased the cost.

Anosov groups: local mixing, counting and equidistribution

Let $G$ be a connected semisimple real algebraic group, and $\Gamma<G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $\langle (\exp tv). f_1, f_2\rangle$ in $L^2(\Gamma\backslash G)$ as $t\to \infty$ for any $f_1, f_2\in C_c(\Gamma\backslash G)$ and any vector $v$ in the interior of the limit cone of $\Gamma$. These asymptotics involve higher rank analogues of Burger-Roblin measures which are introduced in this paper. As an application, for any affine symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$.

Cremona Groups Over Finite Fields, Neretin Groups, and Non-Positively Curved Cube Complexes

Abstract We show that plane Cremona groups over finite fields embed as dense subgroups into Neretin groups, that is, groups of almost automorphisms of rooted trees. We also show that if the finite base field has even characteristic and contains at least four elements, then the permutations induced by birational transformations on rational points of regular projective surfaces are even. In a second part, we construct explicit locally compact CAT(0) cube complexes, on which Neretin groups act properly. This allows us to recover in a unified way various results on Neretin groups such as that they are of type $F_{\infty }$. We also prove a new fixed-point theorem for CAT(0) cube complexes without infinite cubes and use it to deduce a regularization theorem for plane Cremona groups over finite fields.

Zariski density and computing with S-integral groups

We generalize our methodology for computing with Zariski dense subgroups of SL(n,Z) and Sp(n,Z), to accommodate input dense subgroups H of SL(n,Q) and Sp(n,Q). A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of H. Once we have this overgroup, we may describe all congruence quotients of H. The case n=2 receives particular attention.

Local Mixing of One-Parameter Diagonal Flows on Anosov Homogeneous Spaces

Abstract Let $G$ be a connected semisimple real algebraic group and $\Gamma &amp;lt; G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp (t\mathsf {v}): t \in {\mathbb {R}}\}$ on $\Gamma \backslash G$ for any interior direction $\mathsf {v}$ of the limit cone of $\Gamma $ with respect to the Bowen–Margulis–Sullivan measure associated to $\mathsf {v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf {u}$ in some fixed subspace transverse to $\mathsf {v}$. We also obtain a uniform bound for the correlation function, which decays exponentially in $\|\mathsf {u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(\Gamma \backslash G)$ proved by Edwards–Lee–Oh.