Non-virtually abelian anisotropic linear groups are not boundedly generated

We prove that if a linear group $$\Gamma \subset \mathrm {GL}_n(K)$$ Γ ⊂ GL n ( K ) over a field K of characteristic zero is boundedly generated by semi-simple (diagonalizable) elements then it is virtually solvable. As a consequence, one obtains that infinite S -arithmetic subgroups of absolutely almost simple anisotropic algebraic groups over number fields are never boundedly generated. Our proof relies on Laurent’s theorem from Diophantine geometry and properties of generic elements.

[1]  Dong Quan Ngoc Nguyen,et al.  Polynomial Parametrization for SL2 over Quadratic Number Rings , 2018, International Mathematics Research Notices.

[2]  K. Fujiwara.  The second bounded cohomology of an amalgamated free product of groups , 1999 .

[3]  U. Zannier,et al.  Applications of Diophantine Approximation to Integral Points and Transcendence , 2018 .

[4]  Jacques Tits,et al.  Groupes Réductifs , 1965 .

[5]  Gregory Margulis,et al.  Discrete Subgroups of Semisimple Lie Groups , 1991 .

[6]  A. Rapinchuk,et al.  Existence of irreducible R-regular elements in Zariski-dense subgroups , 2003 .

[7]  A. Rapinchuk,et al.  Generic elements in Zariski-dense subgroups and isospectral locally symmetric spaces , 2012, 1212.1217.

[8]  G. Willis,et al.  Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity , 2009, 0911.1966.

[9]  Armand Borel.  Linear Algebraic Groups , 1991 .

[10]  A. Lubotzky.  Subgroup growth and congruence subgroups , 1995 .

[11]  K. Fujiwara.  The Second Bounded Cohomology of a Group Acting on a Gromov‐Hyperbolic Space , 1998 .

[12]  Bounded cohomology of group constructions , 1996 .

[13]  M. Burger,et al.  Bounded cohomology of lattices in higher rank Lie groups , 1999 .

[14]  B. Sury,et al.  Bounded generation of wreath products , 2015 .

[15]  M. Raghunathan.  Discrete subgroups of Lie groups , 1972 .

[16]  G. Keller,et al.  BOUNDED ELEMENTARY GENERATION OF SL,n (0) , 1983 .

[17]  Bounded Generation of S-Arithmetic Subgroups of Isotropic Orthogonal Groups over Number Fields , 2005, math/0508480.

[18]  L. Vaserstein.  Polynomial parametrization for the solutions of Diophantine equations and arithmetic groups , 2010 .

[19]  On uniform exponential growth for linear groups , 2001, math/0108157.

[20]  F. Murnaghan,et al.  LINEAR ALGEBRAIC GROUPS , 2005 .

[21]  Yehuda Shalom,et al.  Bounded generation and Kazhdan’s property (T) , 1999 .

[22]  F. Grunewald,et al.  Free non-abelian quotients of SL2 over orders of imaginary quadratic numberfields , 1981 .

[23]  D. Segal,et al.  Subgroups of finite index in soluble groups: II , 1987 .

[24]  László Pyber,et al.  Bounded Generation and Linear Groups , 2003, Int. J. Algebra Comput..

[25]  B. Sury,et al.  Bounded generation of SL2 over rings of S-integers with infinitely many units , 2017, Algebra & Number Theory.

[26]  H. Bass.  Theorems of Jordan and Burnside for algebraic groups , 1983 .

[27]  Jean-Pierre Serre,et al.  Le Probleme des Groupes de Congruence Pour SL 2 , 1970 .

[28]  Martin Kassabov,et al.  Kazhdan Constants for Sln(Z) , 2005, Int. J. Algebra Comput..

[29]  M. Raghunathan,et al.  The congruence subgroup problem , 2004 .

[30]  A. Rapinchuk,et al.  ABSTRACT PROPERTIES OF $ S$-ARITHMETIC GROUPS AND THE CONGRUENCE PROBLEM , 1993 .

[31]  D. Segal,et al.  Finitely generated groups with polynomial index growth , 2007, 0711.0687.

[32]  A. Rapinchuk,et al.  Generic elements of a Zariski-dense subgroup form an open subset , 2017, 1707.02508.

[33]  A. L. Shmel’kin.  Polycyclic groups , 1968 .

[34]  O. I. Tavgen'.  BOUNDED GENERATION OF CHEVALLEY GROUPS OVER RINGS OF ALGEBRAIC S-INTEGERS , 1991 .

[35]  Yehuda Pinchover,et al.  Geometry, Spectral Theory, Groups, and Dynamics , 2005 .