Word maps have large image

By Michael Larsen. 2002

70 citations recorded by the source corpus.

AbstractAn elementw in the free group onr letters defines a mapfw,G:Gv→G for each groupG. In this note, we show that wheneverw≠1 andG is a semisimple algebraic group,fw,G is dominant. As an application, we show that for fixedw and Γi a sequence of pairwise non-isomorphic finite simple groups, $$\mathop {lim}\limits_{i \to \infty } \frac{{log\left| {\Gamma _j } \right|}}{{log\left| {f_w ,\Gamma _j (\Gamma _j^r )} \right|}} = 1$$ .

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e4515be53f098e2c3ff77f74cefe38fc336cdb30

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Citations in the corpus

On the joint spectral radius for isometries of non-positively curved spaces and uniform growth (2018, unknown): “d R B in O(2n) of Aand Bin such a way that the group they generate is a free group which does not have 1 as an eigenvalue. This fact follows easily from Borel’s theorem on the dominance of words maps [15,59] in simple algebraic groups (here SO(2n)) and from the fact that in even dimension a generic ”
Strongly dense free subgroups of semisimple algebraic groups (2010, background): “See Borel [3] or Larsen [31].”
The images of non-commutative polynomials evaluated on 2 x 2 matrices (2010, background): “p-group is related on the one hand with images of Lie polynomials and words in groups, and on the other hand with problems of Specht type, which is of signiﬁcant current interest. Borel [B] (also cf. [La]) provedthat for any simple (semisimple) algebraicgroupG and any word w of the free group on r v”
Product decompositions of quasirandom groups and a Jordan type theorem (2007, methodology): “The proof of Proposition 1.2 relies on algebraic geometry (replacing [ La ]) and group theoretic arguments.”
Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture (2020, background): “asonable estimate of w(G). The typical classes of groups which provide such estimates are simple algebraic groups, simple and perfect nite groups, and some others. The theorem of Borel [16] (also cf. [70]) states that for any simple (semisimple) algebraic group Gand any word wof the free group on mv”
Word maps and Waring type problems (2007, unknown): “We consider four cases: (1) L(h) = L. (2) L(h) is an inseparable extension of L. (3) L(h) is a non-trivial separable extension of L. (4) L (h) is a transcendental extension of L.”
Expansion in finite simple groups of Lie type (2013, methodology): “(This type of algebro-geometric viewpoint is of the same type as that used in, for example, [44].”
Characters of symmetric groups: sharp bounds and applications (2008, unknown): “See Borel [Bo] for algebraic groups, Nikolov and Segal [NS1], [NS2] for profinite groups, as well as [LiSh1], [DPSSh], [ L ], [Sh] and [LaSh].”
Commutator maps, measure preservation, and T-systems (2007, background): “Word maps on algebraic groups and on finite simple groups have been the subject of active investigations in recent years, see [Bo], [LiSh2], [ La ], [Sh] and [LaSh].”
Word maps, conjugacy classes, and a noncommutative Waring-type theorem (2009, methodology): “In the first stage of the proof we rely on a result of Borel [2] that a word map is a dominant map at the level of simple algebraic groups, and consequences proved by Larsen in [17] showing that word maps have large image in finite simple groups.”