Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, II: groups of type F, G, and H

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as rationality and functional equations. Here, we calculate such bivariate zeta functions of three infinite families of nilpotent groups of class 2 generalizing the Heisenberg group of (3×3)-unitriangular matrices over rings of integers of number fields. The local factors of these zeta functions are also expressed in terms of sums over finite hyperoctahedral groups, which provide formulae for joint distributions of three statistics on such groups

Twisted conjugacy in soluble arithmetic groups

We investigate the ongoing problem of classifying which S-arithmetic groups have the so-called property $R_\infty$. While non-amenable S-arithmetic groups tend to have $R_\infty$, the soluble case seems more delicate. Here we address Borel subgroups in type A and show how the problem reduces to determining whether a metabelian subgroup of $\mathrm{GL}_2$ has $R_\infty$. For higher solubility class we show how automorphisms of the base ring give $R_\infty$. Our results yield many families of soluble S-arithmetic groups with $R_\infty$ but we also exhibit metabelian families not manifesting it. We formulate a conjecture concerning $R_\infty$ for the groups in question, addressing their geometric properties and algebraic structure.Comment: 47 page

Thompson-like groups, Reidemeister numbers, and fixed points

We investigate fixed-point properties of automorphisms of groups similar to R. Thompson's group $F$. Revisiting work of Gon\c{c}alves-Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the abelianization, implying the so-called property $R_\infty$. Using the BNS $\Sigma$-invariant and drawing from works of Gon\c{c}alves-Sankaran-Strebel and Zaremsky, we show that our tool applies to many $F$-like groups, including Stein's $F_{2,3}$, Cleary's $F_\tau$, the Lodha-Moore groups, and the braided version of $F$.Comment: v3: 25 pages, 4 figures; Incorporated referees' suggestions, corrected Proposition 2.7, included new remarks. Final version, to appear in Geometriae Dedicat

Bivariate representation and conjugacy class zeta functions associated to unipotent group schemes, I: arithmetic properties

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equations. Moreover, we show that such bivariate zeta functions specialize to (univariate) class number zeta functions. In case of nilpotency class 2, bivariate representation zeta functions also specialize to (univariate) twist representation zeta functions

Bivariate representation and conjugacy classzeta functions associated to unipotent groupschemes

The main topic of this doctoral thesis is zeta functions of groups. Let G be a unipotent group scheme defined over the ring of integers O of a number field. The group G(O) of O-rational points is a finitely generated torsion-free nilpotent group. We introduce two bivariate zeta functions related to groups of the form G(O): firstly the bivariate representation zeta function of G(O), which enumerates the isomorphism classes of irreducible complex representations of finite dimensions of its congruence quotients, and secondly the bivariate conjugacy class zeta function of G(O), which enumerates the conjugacy classes of each size of its congruence quotients. These zeta functions might be used as tools for understanding another (uni-variate) zeta functions, as they both specialise to class number zeta functions, which enumerate class numbers of the congruence quotients. Additionally, in case of nilpotency class two, bivariate representation zeta functions specialise to twist representation zeta functions, which are zeta functions enumerating the irreducible complex characters of finite dimensions up to tensoring by one-dimensional characters. We show that bivariate representation and bivariate conjugacy class zeta functions satisfy Euler decompositions and that almost all of their Euler factors are rational and satisfy functional equations. We also prove that they converge on some domains of C^2 and, furthermore, their maximal domains of convergence and meromorphic continuation are independent of the number field O considered, up to finitely many local factors. We provide formulae for the bivariate zeta functions of three infinite families of groups of nilpotency class 2 of the form G(O) which generalise the Heisenberg group of 3×3-unitriangular matrices over O. As an application, we establish formulae for the joint distributions of three statistics on finite hyperoctahedral groups

Effect of lung recruitment and titrated Positive End-Expiratory Pressure (PEEP) vs low PEEP on mortality in patients with acute respiratory distress syndrome - A randomized clinical trial

IMPORTANCE: The effects of recruitment maneuvers and positive end-expiratory pressure (PEEP) titration on clinical outcomes in patients with acute respiratory distress syndrome (ARDS) remain uncertain. OBJECTIVE: To determine if lung recruitment associated with PEEP titration according to the best respiratory-system compliance decreases 28-day mortality of patients with moderate to severe ARDS compared with a conventional low-PEEP strategy. DESIGN, SETTING, AND PARTICIPANTS: Multicenter, randomized trial conducted at 120 intensive care units (ICUs) from 9 countries from November 17, 2011, through April 25, 2017, enrolling adults with moderate to severe ARDS. INTERVENTIONS: An experimental strategy with a lung recruitment maneuver and PEEP titration according to the best respiratory-system compliance (n = 501; experimental group) or a control strategy of low PEEP (n = 509). All patients received volume-assist control mode until weaning. MAIN OUTCOMES AND MEASURES: The primary outcome was all-cause mortality until 28 days. Secondary outcomes were length of ICU and hospital stay; ventilator-free days through day 28; pneumothorax requiring drainage within 7 days; barotrauma within 7 days; and ICU, in-hospital, and 6-month mortality. RESULTS: A total of 1010 patients (37.5% female; mean [SD] age, 50.9 [17.4] years) were enrolled and followed up. At 28 days, 277 of 501 patients (55.3%) in the experimental group and 251 of 509 patients (49.3%) in the control group had died (hazard ratio [HR], 1.20; 95% CI, 1.01 to 1.42; P = .041). Compared with the control group, the experimental group strategy increased 6-month mortality (65.3% vs 59.9%; HR, 1.18; 95% CI, 1.01 to 1.38; P = .04), decreased the number of mean ventilator-free days (5.3 vs 6.4; difference, −1.1; 95% CI, −2.1 to −0.1; P = .03), increased the risk of pneumothorax requiring drainage (3.2% vs 1.2%; difference, 2.0%; 95% CI, 0.0% to 4.0%; P = .03), and the risk of barotrauma (5.6% vs 1.6%; difference, 4.0%; 95% CI, 1.5% to 6.5%; P = .001). There were no significant differences in the length of ICU stay, length of hospital stay, ICU mortality, and in-hospital mortality. CONCLUSIONS AND RELEVANCE: In patients with moderate to severe ARDS, a strategy with lung recruitment and titrated PEEP compared with low PEEP increased 28-day all-cause mortality. These findings do not support the routine use of lung recruitment maneuver and PEEP titration in these patients. TRIAL REGISTRATION: clinicaltrials.gov Identifier: NCT01374022