Cocompact lattices in locally pro-$p$-complete rank-2 Kac-Moody groups

Capa

Citar

Texto integral

Acesso aberto Acesso aberto
Acesso é fechado Acesso está concedido
Acesso é fechado Somente assinantes

Resumo

We initiate an investigation of lattices in a new class of locally compact groups: so-called locally pro-$p$-complete Kac-Moody groups. We discover that in rank 2 their cocompact lattices are particularly well-behaved: under mild assumptions, a cocompact lattice in this completion contains no elements of order $p$. This statement is still an open question for the Caprace-Remy-Ronan completion. Using this, modulo results of Capdeboscq and Thomas, we classify edge-transitive cocompact lattices and describe a cocompact lattice of minimal covolume. Bibliography: 22 titles.

Sobre autores

Inna Capdeboscq

University of Warwick, Mathematics Institute

Katerina Hristova

School of Mathematics, University of East Anglia

Dmitriy Rumynin

University of Warwick, Mathematics Institute; Laboratory of algebraic geometry and its applications, National Research University "Higher School of Economics" (HSE)

Bibliografia

  1. H. Bass, A. Lubotzky, Tree lattices, Progr. Math., 176, Birkhäuser Boston, Inc., Boston, MA, 2001, xiv+233 pp.
  2. A. Borel, G. Harder, “Existence of discrete cocompact subgroups of reductive groups over local fields”, J. Reine Angew. Math., 298 (1978), 53–64
  3. I. Capdeboscq, K. Kirkina, D. Rumynin, “Presentations of affine Kac–Moody groups”, Forum Math. Sigma, 6:e21 (2018), 35 pp.
  4. I. Capdeboscq, B. Remy, “On some pro-$p$ groups from infinite-dimensional Lie theory”, Math. Z., 278:1-2 (2014), 39–54
  5. I. Capdeboscq, D. Rumynin, “Kac–Moody groups and completions”, J. Algebra (to appear)
  6. I. Capdeboscq, A. Thomas, “Lattices in complete rank 2 Kac–Moody groups”, J. Pure Appl. Algebra, 216:6 (2012), 1348–1371
  7. I. Capdeboscq, A. Thomas, “Co-compact lattices in complete Kac–Moody groups with Weyl group right-angled or a free product of spherical special subgroups”, Math. Res. Lett., 20:2 (2013), 339–358
  8. P.-E. Caprace, N. Monod, “A lattice in more than two Kac–Moody groups is arithmetic”, Israel J. Math., 190 (2012), 413–444
  9. P.-E. Caprace, B. Remy, “Simplicity and superrigidity of twin building lattices”, Invent. Math., 176:1 (2009), 169–221
  10. L. Carbone, M. Ershov, G. Ritter, “Abstract simplicity of complete Kac–Moody groups over finite fields”, J. Pure Appl. Algebra, 212:10 (2008), 2147–2162
  11. L. Carbone, H. Garland, “Existence of lattices in Kac–Moody groups over finite fields”, Commun. Contemp. Math., 5:5 (2003), 813–867
  12. R. W. Carter, Y. Chen, “Automorphisms of affine Kac–Moody groups and related Chevalley groups over rings”, J. Algebra, 155:1 (1993), 44–54
  13. И. М. Гельфанд, М. И. Граев, И. И. Пятецкий-Шапиро, Теория представлений и автоморфные функции, Обобщенные функции, 6, Наука, М., 1966, 512 с.
  14. W. Herfort, L. Ribes, “Torsion elements and centralizers in free products of profinite groups”, J. Reine Angew. Math., 1985:358 (1985), 155–161
  15. Э. Хьюитт, К. Росс, Абстрактный гармонический анализ, т. I, Наука, М., 1975, 654 с.
  16. A. Lubotzky, “Lattices of minimal covolume in $operatorname{SL}_2$: a nonarchimedean analogue of Siegel's theorem $mugeq pi/21$”, J. Amer. Math. Soc., 3:4 (1990), 961–975
  17. T. Marquis, “Around the Lie correspondence for complete Kac–Moody groups and Gabber–Kac simplicity”, Ann. Inst. Fourier (Grenoble), 69:6 (2019), 2519–2576
  18. L. Ribes, P. Zalesskii, Profinite groups, Ergeb. Math. Grenzgeb. (3), 40, Springer-Verlag, Berlin, 2000, xiv+435 pp.
  19. G. Rousseau, “Groupes de Kac–Moody deployes sur un corps local II. Masures ordonnees”, Bull. Soc. Math. France, 144:4 (2016), 613–692
  20. C. L. Siegel, “Some remarks on discontinuous groups”, Ann. of Math. (2), 46:4 (1945), 708–718
  21. J. Tits, “Uniqueness and presentation of Kac–Moody groups over fields”, J. Algebra, 105:2 (1987), 542–573
  22. J. Tits, “Ensembles ordonnes, immeubles et sommes amalgamees”, Bull. Soc. Math. Belg. Ser. A, 38 (1986), 367–387

Arquivos suplementares

Arquivos suplementares
Ação
1. JATS XML
Baixar

Declaração de direitos autorais © Capdeboscq I., Hristova K., Rumynin D.A., 2020

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).