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Journal of Lie Theory 26 (2016), No. 4, 1079--1105
Copyright Heldermann Verlag 2016



Isomorphisms and Rigidity of Arithmetic Kac-Moody Groups

Amir Farahmand Parsa
Mathematisches Institut, Justus-Liebig-Universitдt, Arndtstrasse 2, 35392 Giessen, Germany
[email protected]

Max Horn
Mathematisches Institut, Justus-Liebig-Universitдt, Arndtstrasse 2, 35392 Giessen, Germany
[email protected]

Ralf Kцhl
Mathematisches Institut, Justus-Liebig-Universitдt, Arndtstrasse 2, 35392 Giessen, Germany
[email protected]



We solve the isomorphism problem for subgroups of integral points of two-spherical Kac-Moody groups over the rational numbers. Along the way we establish versions of Mostow-Margulis strong rigidity and Margulis superrigidity with target in two-spherical split Kac-Moody groups over the rational numbers for arithmetically defined subgroups.

Keywords: Arithmetic Kac-Moody group, twin building, isomorphism problem, Mostow-Margulis strong rigidity, Margulis superrigidity.

MSC: 20G44, 20G25, 51E24

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AuthorВ PopulationВ Age, mean В± SD (years); male sex (%)В Inclusion criteriaВ Study designВ Modality (parameter)В OutcomeВ 
London et al. [68] 92 HD patients 49.9 ± 15.9; 52 •Not given •Cross-sectional •Doppler (cfPWV) Echo (LVM) •LVM was increased in HD patients (246 ± 56 versus 198.4 ± 52 g, P = 0.0001) and correlated with aPWV (r = 0.576, P < 0.0001) 
90 controlsВ 50.8 В± 15.8В 
London et al. [96] 138 HD patients Responders: 48.2 ± 14.4; 60 Non-responders: 53.2 ± 17; 53 •HD ≥ 3 months, pre-dialysis BP > 160/90, good quality echocardiography, follow-up  ≥ 9 months •Observational, 4.8-year mean follow-up •Doppler (cfPWV) Echo (LVM) •‘Responders’ were those whose cfPWV decreased in response to treatment. Decreased cfPWV correlated with reduced LVMI (r = 0.566, P < 0.001). Changes in cfPWV and LVM were independently correlated with serum CRP (P < 0.001) 
Nitta et al. [11] 49 HD patients 60.4 ± 1.6, 55 •HD ≥ 6 months •Cross-sectional •Mechano-transducer (brachial and tibial PWV) Echo (LVMI) •LVMI correlated with PWV (r = 0.439, P = 0.001) 
Kim et al. [97] 391 incident HD patients 54.7 ± 13.2; 59 •HD patients: Age ≥18 years, enrolled within 6 months of HD initiation •Cross-sectional •Applanation tonometry (cfPWV) Echo (LVMI) •Univariate regression (a) and multivariate regression (b) showed no significant relationship between PWV and LVMI: (a) β = −0.42 (−1.78, 0.94), P = 0.55; (b) β = 0.19 (−1.41, 1.79), P = 0.82. 
Edwards et al. [34] 117 patients with Stage 2–3 CKD CKD 2: 55.9 ± 11.6; 50 CKD Stage 3: 53.8 ± 11.8; 68 •18–80 years, Stage 2 or 3 CKD. No overt CVD, DM or PVD •Cross-sectional •1.5T CMR (AD and LVM) •LVM was inversely correlated with AD (r = −0.284, P < 0.001) 
40 controlsВ 50.3 В± 9.2; 50В 
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Xvideos, It was a mystical city in which the imagination faltered like one who steps out of the light into darkness; the soul walked naked to and fro, knowing the unknowable, and conscious strangely of experience, intimate but inexpressible, of the absolute. The new ruler, Louis xvideo xx 661f was only a child, and the management of affairs for a second period of eighteen years passed into the hands of Cardinal Mazarin.

Xvideos, It was a mystical city in which the imagination faltered like one who steps out of the light into darkness; the soul walked naked to and fro, knowing the unknowable, and conscious strangely of experience, intimate but inexpressible, of the absolute. The new ruler, Louis xvideo xx 661f was only a child, and the management of affairs for a second period of eighteen years passed into the hands of Cardinal Mazarin.



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