When is a map given by a word surjective? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T01:14:09Z http://mathoverflow.net/feeds/question/735 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/735/when-is-a-map-given-by-a-word-surjective When is a map given by a word surjective? H A Helfgott 2009-10-16T11:08:59Z 2009-11-02T15:38:11Z <p>Let w(x,y) be a word in x and y. </p> <p>Let x and y now vary in SL_n(K), where K is a field. (Assume, if you wish, that K is an algebraically complete field of characteristic bigger than a constant.)</p> <p>I would like to know for which words w the map</p> <p>y -> w(x,y)</p> <p>isn't surjective (or even dominant - that is, "almost surjective") for x generic.</p> <p>It is clear, for example, that the map is surjective for w(x,y)=xy, and that it isn't surjective for w(x,y)= y x y^{-1}, or for w(x,y) = y x^n y^{-1}, n an integer: all elements of the image of y -> y x^n y^{-1} lie in the same conjugacy class. A moment's thought (thanks, Philipp!) shows that w(x,y) = x y x^n y^{-1} isn't surjective either: its image is just x* im(y->y x^n y^{-1}), and, as we just said, y-> y x^n y^{-1} isn't surjective.</p> <p>I would like to know if the only words w for which the map isn't surjective for x generic are the w's of the form w(x,y) = x^a v(x,y) x^b (v(x,y))^{-1} x^c, where v is some word and a,b,c are some integers. (This seems to me a sensible guess, though I would actually be quite glad if it weren't true.)</p> http://mathoverflow.net/questions/735/when-is-a-map-given-by-a-word-surjective/741#741 Answer by David Speyer for When is a map given by a word surjective? David Speyer 2009-10-16T14:08:20Z 2009-10-16T14:08:20Z <p>This paper <a href="http://front.math.ucdavis.edu/math.GR/0211302" rel="nofollow">http://front.math.ucdavis.edu/math.GR/0211302</a> seems related, although it asks a slightly different question.</p> http://mathoverflow.net/questions/735/when-is-a-map-given-by-a-word-surjective/772#772 Answer by Hugh Thomas for When is a map given by a word surjective? Hugh Thomas 2009-10-16T18:04:36Z 2009-10-16T18:04:36Z <p>I'm going to say some things which might be either (a) obvious, (b) wrong, or (c) useless. (Or some combination!)</p> <p>You could rephrase the question by asking that the map from SL_k x SL_k to SL_k x SL_k given by (x,y) -> (x,w(x,y)) is dominant. This seems like an improvement, because now you're talking about a map between two spaces of equal dimension. </p> <p>If the corresponding map on the tangent spaces of Id x Id is an isomorphism, then certainly the map is dominant. (And this map is easy to compute, given w -- we just replace the product in w by the corresponding sum of tangent vectors.)</p> <p>The map is dominant iff the map on tangent spaces is generically an isomorphism. I don't know how to check this, but my feeling is that it will be easier for this to fail than in the given conjecture. </p> http://mathoverflow.net/questions/735/when-is-a-map-given-by-a-word-surjective/848#848 Answer by Philipp Lampe for When is a map given by a word surjective? Philipp Lampe 2009-10-17T08:25:18Z 2009-10-17T08:25:18Z <p>At least for n=2, the map y -> xyx<sup>-1y<sup>-1</sup> is not surjective for generic x.</p> <p>Let us prove that the map is not surjective for diagonalizable x. Thus, it is not surjective for generic x.</p> <p>Suppose that x is diagonalizable and let a=(a<sub>ij</sub>) be matrix in SL<sub>2</sub>(K). We want to solve the equation xyx<sup>-1</sup>y<sup>-1</sup>=a. By conjugating with an appropriate matrix, we may wlog assume that x=diag(b,b<sup>-1</sup>) is a diagonal matrix. Let y=(y<sub>ij</sub>). A short calculation shows that the diagonal entries in the matrix xyx<sup>-1</sup>y<sup>-1</sup> are</p> <p>a<sub>11</sub>=y<sub>11</sub>y<sub>22</sub>-b<sup>2</sup>y<sub>12</sub>y21</sup>=1+(1-b<sup>2</sup>)y<sub>12</sub>y<sub>21</sub>, a<sub>22</sub>=y<sub>11</sub>y<sub>22</sub>-b<sup>-2</sup>y<sub>12</sub>y<sub>21</sub>=1+(1-b<sup>-2</sup>)y<sub>12</sub>y<sub>21</sub>.</p> <p>This means that in the image there are only matrices a with (a<sub>11</sub>-1)/(a<sub>22</sub>-1)=-b<sup>2</sup>.</p> <p>But xyx<sup>-1</sup>y<sup>-1</sup> has not the form vx<sup>k</sup>v<sup>-1</sup>.</p> http://mathoverflow.net/questions/735/when-is-a-map-given-by-a-word-surjective/3807#3807 Answer by H A Helfgott for When is a map given by a word surjective? H A Helfgott 2009-11-02T15:38:11Z 2009-11-02T15:38:11Z <p>The discussion has now moved to</p> <p><a href="http://mathoverflow.net/questions/2082/surjective-maps-given-by-words-redux" rel="nofollow">http://mathoverflow.net/questions/2082/surjective-maps-given-by-words-redux</a></p>