Free Subgroups of Linear Groups

By G. Soifer. 2007

430 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6e9114a59fcb8eb5b38aa822463b71849f28976f

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Strongly dense free subgroups of semisimple algebraic groups (2010, background): “as a trivial consequence of the Tits alternative [48], according to which every finitely generated subgroup of invertible linear transformations over a field contains a non abelian free subgroup or a solvable subgroup of finite index.”
Orbit decidability and the conjugacy problem for some extensions of groups (2007, background): “Tits [40] proved the deep and remarkable fact that every finitely generated subgroup of GLn(Z) is either virtually solvable or it contains a non-abelian free subgroup.”
Groups acting on manifolds: around the Zimmer program (2008, background): “iately implies that actions of the type described cannot occur on manifolds with amenable fundamental group or even on manifolds whose fundamental groups do not contain a free group on two generators [223]. Expository accounts of the proof can be found in [66, 252, 255]. The representation ρis actua”
A Spectral Gap Theorem in $SU(d)$ (2011, background): “ of Q. We will consider V and Las vector spaces over k, hence V = {g∈ Matd×d(k),Trg= 0}. (i) Exploiting the theory of random matrix products requires proximal elements. Following the approach of Tits [Tits] (see also [G]) proximal elements in a suitable setting may be produced by passing to an appro”
Milnor's Problem on the Growth of Groups and its Consequences (2011, background): “a subgroup of GLn(F), n ≥ 1, F a field), either contains a free subgroup on two generators or is virtually solvable [210].”
Locally Homogeneous Geometric Manifolds (2010, methodology): “Using the Tits Alternative [162], he reduced this question to whether a rank two free group could act properly by affine transformations on R.”
Approximate Subgroups of Linear Groups (2010, methodology): “As a straightforward corollary we obtain the following special case of Gromov’s theorem [31], first established by combining the Tits Alternative [59] with a theorem of Milnor [45] and Wolf [64].”
Affine linear sieve, expanders, and sum-product (2010, background): “However as far as upper bounds go this says nothing about the original orbit since after applying Tits Theorem is possibly of infinite index in .”
Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (2010, background): “1 comes from linear groups and the Tits alternative (see [55]), which was originally formulated for finitely generated linear groups:”
Combinatorial algebra : syntax and semantics (2014, neutral): “By the Tits alternative [313], every finitely generated group of matrices over a field either satisfies no nontrivial identity or has a solvable subgroup of finite index (the condition “finitely generated” can be removed if the characteristic is 0).”
Isometry groups of non‐positively curved spaces: discrete subgroups (2009, background): “By Tits’ alternative [86], the amenable radical of Λ is soluble-by-locally-finite and thus locally finite since τ(Λ) is Zariski-dense and H is semi-simple.”
C*-simplicity and the unique trace property for discrete groups (2014, background): “As in the proof of the Tits alternative [51], or as in [7], passing to a finite index subgroup if necessary, and looking at various linear representations, one can let the linear group G act proximally and strongly irreducibly on projective spaces over local fields.”