Open Subgroup Research Articles

Densely locally minimal groups

We study locally compact groups having all dense subgroups (locally) minimal. We call such groups densely (locally) minimal. In 1972 Prodanov proved that the infinite compact abelian groups having all subgroups minimal are precisely the groups Zp of p-adic integers. In [30], we extended Prodanov's theorem to the non-abelian case at several levels. In this paper, we focus on the densely (locally) minimal abelian groups.We prove that in case that a topological abelian group G is either compact or connected locally compact, then G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Zp for some prime p. This should be compared with the main result of [9]. Our Theorem C provides another extension of Prodanov's theorem: an infinite locally compact group is densely minimal if and only if it is isomorphic to Zp. In contrast, we show that there exists a densely minimal, compact, two-step nilpotent group that neither is a Lie group nor it has an open subgroup isomorphic to Zp.

Predecessors of topologies of LCA groups

We show in this paper that for a non-compact LCA group (G,τ), τ has exactly 22k(G) predecessors in G2(G). This answers a problem posed in the literature affirmatively. Denote by N the class of all LCA groups (G,τ) such that P2(τ) is precompact. It is shown that for an LCA group G, G∈N if and only if G≅Rn×H, where n is a non-negative integer and H is an LCA group with an open compact subgroup N such that H/N∈M. As an application of this result, we extend a well known result on discrete abelian groups to the case of LCA groups. We show that if (G,τ) is a non-compact LCA group and σ is a predecessor of τ in G2(G), then the connected component of (G,τ) coincides with the connected component of (G,σ). It is also shown that for a non-compact LCA group (G,τ), if σ is a predecessor of τ in G2(G), then the equality ib(G,τ)=ib(G,σ) holds. This partially answer a question posed in the literature.

A Comparative Analysis of Physiotherapy for Stress Urinary Incontinence after Open or Robotic-Assisted Radical Prostatectomy.

Purpose: We determined whether the patient-reported benefits of physiotherapy for stress urinary incontinence (SUI) symptoms differ significantly between men who have had open prostatectomy and those who have had robotic-assisted laparoscopic prostatectomy. Method: We conducted a retrospective analysis of data collected from the Rapid Access Clinic 4 offered by the Prostate Cancer Centre in Calgary, Alberta. Baseline characteristics were measured at the pre-surgery appointment, including demographics, health factors, and potential risk factors for SUI. Patient-reported SUI symptoms were measured pre- and post-surgery using the global score of the International Consultation on Incontinence Questionnaire-Urinary Incontinence. Results: Data from 56 men were included in the analysis, evenly split between the open and robotic sub-groups. At 3 months post-surgery, no statistically significant differences were found in the factors associated with incontinence between the two sub-groups. There was a statistically significant improvement in self-reported incontinence symptom severity from 3-month to 2-year follow-up for both sub-groups. Physiotherapy did not differentially affect either sub-group in a significant way. Conclusions: The self-reported benefits of physiotherapy for SUI symptoms did not differ significantly between the two types of prostatectomy surgery at 2 years post-surgery.

DISTALITY OF CERTAIN ACTIONS ON -ADIC SPHERES

Consider the action of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on the $p$-adic unit sphere ${\mathcal{S}}_{n}$ arising from the linear action on $\mathbb{Q}_{p}^{n}\setminus \{0\}$. We show that for the action of a semigroup $\mathfrak{S}$ of $\operatorname{GL}(n,\mathbb{Q}_{p})$ on ${\mathcal{S}}_{n}$, the following are equivalent: (1) $\mathfrak{S}$ acts distally on ${\mathcal{S}}_{n}$; (2) the closure of the image of $\mathfrak{S}$ in $\operatorname{PGL}(n,\mathbb{Q}_{p})$ is a compact group. On ${\mathcal{S}}_{n}$, we consider the ‘affine’ maps $\overline{T}_{a}$ corresponding to $T$ in $\operatorname{GL}(n,\mathbb{Q}_{p})$ and a nonzero $a$ in $\mathbb{Q}_{p}^{n}$ satisfying $\Vert T^{-1}(a)\Vert _{p}<1$. We show that there exists a compact open subgroup $V$, which depends on $T$, such that $\overline{T}_{a}$ is distal for every nonzero $a\in V$ if and only if $T$ acts distally on ${\mathcal{S}}_{n}$. The dynamics of ‘affine’ maps on $p$-adic unit spheres is quite different from that on the real unit spheres.

Urban Open Space Is Associated with Better Renal Function of Adult Residents in New Taipei City.

The purpose of this study is to explore the association between proximity to open space and adult renal function. This was a cross-sectional study. Adult residents of Taipei metropolis were recruited in the analysis. The proximity of each subject to open space was measured using the Geographic Information System. Residents were divided into two groups: with and without chronic kidney disease (CKD). We made univariable comparisons between the two groups. The logistic regression models were used to estimate the odds ratio of CKD. Forest plot was used to examine the effect of interaction between distance to open space and subgroup variable on CKD. A total number of 21,656 subjects with mean age 53.6 years were enrolled in the study. Of the subjects, 2226 (10.28%) had CKD. The mean and standard deviation of distance to open space were 117.23 m and 80.19 m, respectively. Every 100 m distance to open space was associated with an odds ratio of 1.071 for CKD. Subgroup analysis revealed that residents of female, without hypertension, or without impaired fasting glucose (IFG) living more than 200 m from open spaces have greater odds of CKD than those living less than 200 m. Conclusions: Proximity to open space was associated with a lower prevalence of CKD among adults in Taiwan. Such association was enhanced among females and healthy adults without hypertension or impaired fasting glucose (IFG).

Frames of Translates for Number-Theoretic Groups

Frames of translates of $$f \in L^2(G)$$ are characterized in terms of the zero-set of the so-called spectral symbol of f in the setting of a locally compact abelian group G having a compact open subgroup H. We refer to such a G as a number-theoretic group. This characterization was first proved in 1992 by Li and one of the authors for $$L^2({\mathbb {R}}^d)$$ with the same formal statement of the characterization. For number-theoretic groups, and these include local fields, the strategy of proof is necessarily entirely different, and it requires a new notion of translation that reduces to the usual definition in $${\mathbb {R}}^d$$ .

Accuracy of the NSQIP risk calculator for predicting complications following adrenalectomy.

Adrenalectomy is performed to treat functional pathology and remove tumors of malignant concern. The National Surgical Quality Improvement Program (NSQIP) risk calculator predicts 30-day complications and length of stay following index surgical procedures. We assess whether this tool accurately predicts complications following adrenalectomy procedures at a tertiary care academic medical center. A retrospective review was performed for all adrenalectomies at a single institution from 2004 to 2016. 197 patients underwent adrenalectomy without concurrent resections. Predicted risk for NSQIP complications was calculated for each patient. The mean predicted and observed risks (%) at 30days across all patients within each category were determined, and these were compared with two-sided one-sample t tests. Of 197 adrenalectomies, 180 were laparoscopic and 17 were open. For laparoscopic adrenalectomy, ten (5.5%) complications were observed including nine (5%) graded Clavien III or greater. All observed complication rates were significantly different than predicted (p values for all < 0.005). Mean observed length of stay was also significantly less than predicted (1.6 versus 2.1days, p < 0.001). In the open adrenalectomy subgroup, there were no observed complications with observed mean length of stay equivalent to predicted (5.8 versus 5.3, p = 0.08) without a higher readmission rate (5.9 versus 6.0%). Statistical differences were noted between the actual complication rates of adrenalectomy versus those predicted by the NSQIP calculator. Certain observed differences may not necessarily have clinical significance. Urology procedure-specific calculators may better refine predictions for sub-specialty procedures with future work requisite to determine performance across all practice settings.

Algebraic entropy on strongly compactly covered groups

We introduce a new class of locally compact groups, namely the strongly compactly covered groups, which are the Hausdorff topological groups G such that every element of G is contained in a compact open normal subgroup of G. For continuous endomorphisms ϕ:G→G of these groups we compute the algebraic entropy and study its properties. Also an Addition Theorem is available under suitable conditions.

An induction theorem for groups acting on trees

If G G is a group acting on a locally finite tree X X , and S \mathscr {S} is a G G -equivariant sheaf of vector spaces on X X , then its compactly-supported cohomology is a representation of G G . Under a finiteness hypothesis, we prove that if H c 0 ( X , S ) H_c^0(X, \mathscr {S}) is an irreducible representation of G G , then H c 0 ( X , S ) H_c^0(X, \mathscr {S}) arises by induction from a vertex or edge stabilizing subgroup. If G \boldsymbol {\mathrm {G}} is a reductive group over a nonarchimedean local field F F , then Schneider and Stuhler realize every irreducible supercuspidal representation of G = G ( F ) G = \boldsymbol {\mathrm {G}}(F) in the degree-zero cohomology of a G G -equivariant sheaf on its reduced Bruhat-Tits building X X . When the derived subgroup of G \boldsymbol {\mathrm {G}} has relative rank one, X X is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.

Finite Homomorphic Images of Groups of Finite Rank

Let π be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic π-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-π-group of finite rank has an open normal pronilpotent subgroup.

Hausdorff dimensions in p-adic analytic groups

Let G be a finitely generated pro-p group, equipped with the p-power series. The associated metric and Hausdorff dimension function give rise to the Hausdorff spectrum, which consists of the Hausdorff dimensions of closed subgroups of G. In the case where G is p-adic analytic, the Hausdorff dimension function is well understood; in particular, the Hausdorff spectrum consists of finitely many rational numbers closely linked to the analytic dimensions of subgroups of G. Conversely, it is a long-standing open question whether the finiteness of the Hausdorff spectrum implies that G is p-adic analytic. We prove that the answer is yes, in a strong sense, under the extra condition that G is soluble. Furthermore, we explore the problem and related questions also for other filtration series, such as the lower p-series, the Frattini series, the modular dimension subgroup series and quite general filtration series. For instance, we prove, for odd primes p, that every countably based pro-p group G with an open subgroup mapping onto 2 copies of the p-adic integers admits a filtration series such that the corresponding Hausdorff spectrum contains an infinite real interval.

HYPERREFLEXIVITY CONSTANTS OF THE BOUNDED -COCYCLE SPACES OF GROUP ALGEBRAS AND C*-ALGEBRAS

We introduce the concept of strong property $(\mathbb{B})$ with a constant for Banach algebras and, by applying a certain analysis on the Fourier algebra of the unit circle, we show that all C*-algebras and group algebras have the strong property $(\mathbb{B})$ with a constant given by $288\unicode[STIX]{x1D70B}(1+\sqrt{2})$. We then use this result to find a concrete upper bound for the hyperreflexivity constant of ${\mathcal{Z}}^{n}(A,X)$, the space of bounded $n$-cocycles from $A$ into $X$, where $A$ is a C*-algebra or the group algebra of a group with an open subgroup of polynomial growth and $X$ is a Banach $A$-bimodule for which ${\mathcal{H}}^{n+1}(A,X)$ is a Banach space. As another application, we show that for a locally compact amenable group $G$ and $1&lt;p&lt;\infty$, the space $CV_{P}(G)$ of convolution operators on $L^{p}(G)$ is hyperreflexive with a constant given by $384\unicode[STIX]{x1D70B}^{2}(1+\sqrt{2})$. This is the generalization of a well-known result of Christensen [‘Extensions of derivations. II’, Math. Scand. 50(1) (1982), 111–122] for $p=2$.

Zeta functions associated to admissible representations of compact 𝑝-adic Lie groups

Let G G be a profinite group. A strongly admissible smooth representation ϱ \varrho of G G over C {\mathbb {C}} decomposes as a direct sum ϱ ≅ ⨁ π ∈ Irr ( G ) m π ( ϱ ) π \varrho \cong \bigoplus _{\pi \in \text {Irr}(G)} m_\pi (\varrho ) \, \pi of irreducible representations with finite multiplicities m π ( ϱ ) m_\pi (\varrho ) such that, for every positive integer n n , the number r n ( ϱ ) r_n(\varrho ) of irreducible constituents of dimension n n is finite. Examples arise naturally in the representation theory of reductive groups over nonarchimedean local fields. In this article we initiate an investigation of the Dirichlet generating function \[ ζ ϱ ( s ) = ∑ n = 1 ∞ r n ( ϱ ) n − s = ∑ π ∈ Irr ( G ) m π ( ϱ ) ( dim ⁡ π ) s \zeta _\varrho (s) = \sum \nolimits _{n=1}^\infty r_n(\varrho ) n^{-s} = \sum \nolimits _{\pi \in \text {Irr}(G)} \frac {m_\pi (\varrho )}{(\dim \pi )^s} \] associated to such a representation ϱ \varrho . Our primary focus is on representations ϱ = Ind H G ( σ ) \varrho = \text {Ind}_H^G(\sigma ) of compact p p -adic Lie groups G G that arise from finite-dimensional representations σ \sigma of closed subgroups H H via the induction functor. In addition to a series of foundational results—including a description in terms of p p -adic integrals—we establish rationality results and functional equations for zeta functions of globally defined families of induced representations of potent pro- p p groups. A key ingredient of our proof is Hironaka’s resolution of singularities, which yields formulae of Denef type for the relevant zeta functions. In some detail we consider representations of open compact subgroups of reductive p p -adic groups that are induced from parabolic subgroups. Explicit computations are carried out by means of complementing techniques: (i) geometric methods that are applicable via distance-transitive actions on spherically homogeneous rooted trees, and (ii) the p p -adic Kirillov orbit method. Approach (i) is closely related to the notion of Gelfand pairs and works equally well in positive defining characteristic.

Profinite groups in which centralizers are abelian

The article deals with profinite groups in which the centralizers are abelian (CA-groups), that is, with profinite commutativity-transitive groups. It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CA-group. It is shown that G has a normal open subgroup N which is either abelian or pro-p. Further, rather detailed information about the finite quotient G/N is obtained.

Intersections of maximal subgroups in prosolvable groups

Given a positive real number we say that an open subgroup H of a profinite group G is an η-intersection if there exists a family of maximal subgroups such that and . We study the finitely generated prosolvable groups G with the property that there exists η such that every maximal intersection in G is an η-intersection. We prove in particular that this is equivalent to a condition on the centralizers in G of the complemented chief factors of G. We use the notion of η-intersection to address an open question asking whether the number of maximal intersections of index n in a finitely generated prosolvable group G is polynomially bounded. We prove that the answer is positive if the derived subgroup of G is pronilpotent.

Homomorphisms into totally disconnected, locally compact groups with dense image

Abstract Let ϕ : G → H {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair ( G , ϕ - 1 ⁢ ( L ) ) {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

Induction for locally compact quantum groups revisited

AbstractIn this paper, we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

Approximating Simple Locally Compact Groups by Their Dense Locally Compact Subgroups

Abstract The class $\mathscr{S}$ of totally disconnected locally compact (tdlc) groups that are non-discrete, compactly generated, and topologically simple contains many compelling examples. In recent years, a general theory for these groups, which studies the interaction between the compact open subgroups and the global structure, has emerged. In this article, we study the non-discrete tdlc groups $H$ that admit a continuous embedding with dense image into some $G\in \mathscr{S}$; that is, we consider the dense locally compact subgroups of groups $G\in \mathscr{S}$. We identify a class $\mathscr{R}$ of almost simple groups that properly contains $\mathscr{S}$ and is moreover stable under passing to a non-discrete dense locally compact subgroup. We show that $\mathscr{R}$ enjoys many of the same properties previously obtained for $\mathscr{S}$ and establish various original results for $\mathscr{R}$ that are also new for the subclass $\mathscr{S}$, notably concerning the structure of the local Sylow subgroups and the full automorphism group.

Open subgroups of the automorphism group of a right-angled building

We study the group of type-preserving automorphisms of a right-angled building, in particular when the building is locally finite. Our aim is to characterize the proper open subgroups as the finite index closed subgroups of the stabilizers of proper residues. One of the main tools is the new notion of firm elements in a right-angled Coxeter group, which are those elements for which the final letter in each reduced representation is the same. We also introduce the related notions of firmness for arbitrary elements of such a Coxeter group and n-flexibility of chambers in a right-angled building. These notions and their properties are used to determine the set of chambers fixed by the fixator of a ball. Our main result is obtained by combining these facts with ideas by Pierre-Emmanuel Caprace and Timothee Marquis in the context of Kac–Moody groups over finite fields, where we had to replace the notion of root groups by a new notion of root wing groups.

Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

A one-to-one correspondence between shifts of group-like projections on a locally compact quantum group [Formula: see text] which are preserved by the scaling group and contractive idempotent functionals on the dual [Formula: see text] is established. This is a generalization of the Illie–Spronk’s correspondence between contractive idempotents in the Fourier–Stieltjes algebra of a locally compact group [Formula: see text] and cosets of open subgroups of [Formula: see text]. We also establish a one-to-one correspondence between nondegenerate, integrable, [Formula: see text]-invariant ternary rings of operators [Formula: see text], preserved by the scaling group and contractive idempotent functionals on [Formula: see text]. Using our results, we characterize coideals in [Formula: see text] admitting an atom preserved by the scaling group in terms of idempotent states on [Formula: see text]. We also establish a one-to-one correspondence between integrable coideals in [Formula: see text] and group-like projections in [Formula: see text] satisfying an extra mild condition. Exploiting this correspondence, we give examples of group-like projections which are not preserved by the scaling group.