| Summary: | <p>Given a group word <em>w</em> in <em>k</em> variables, a group <em>GM</em> and <em>g</em> &in; <em>G</em>, we consider the set <em>S<sub>w</sub></em>(<em>g</em>) of <em>k</em>-tuples (<em>g<sub>1</sub></em>,..., <em>g<sub>k</sub></em>) &in; <em>G</em><sup>(<em>k</em>)</sup> such that <em>w</em>(<em>g<sub>1</sub></em>,..., <em>g<sub>k</sub></em>) = <em>g</em> and when <em>G</em> is finite, the size of <em>S<sub>w</sub></em>(<em>g</em>), <em>N<sub>w</sub></em>(<em>g</em>). N. Amit conjectured that for any finite nilpotent group <em>G</em> and any word in <em>k</em> variables, <em>N<sub>w</sub></em>(1) ≥ |<em>G</em>|<sup><em>k</em>-1</sup>. In this thesis we first prove Amit’s conjecture for finite groups of nilpotency class 2. This was independently proved by Levy in [1]. More generally, we study the class functions <em>N<sub>w</sub></em> for this class of groups and show that the inequality can be improved to <em>N<sub>w</sub></em>(1) ≥ |<em>G</em>|<sup><em>k</em></sup>/|<em>G<sub>w</sub></em> (<em>G<sub>w</sub></em> is the set of <em>w</em>-values in <em>G</em>) if <em>G</em> has odd order. This last result is explained by the fact that the functions <em>N<sub>w</sub></em> are characters of <em>G</em> in this case. For groups of even order, all that can be said is that <em>N<sub>w</sub></em> is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when <em>N<sub>x</sub></em><sup>n</sup></p> is a character if <em>G</em> is a 2-group of nilpotency class 2. We also address the (much harder) problem of studying if <em>N<sub>w</sub></em>(<em>g</em>) ≥ |<em>G</em>|<em><sup>k-1</sup></em> for <em>g</em> &in; <em>G<sub>w</sub></em>, proving that this is the case for the free <em>p</em>-groups of nilpotency class 2 and exponent <em>p</em>. <p>Finally, we look at the analogous problem for finitely generated pro-p groups. Let <em>G</em> be a finitely generated pro-<em>p</em> group and {<em>G<sub>n</sub></em>} some filtration. We define the dimension of a closed subset <em>H</em> ⊆ <em>G</em> as</p> <p> <table align="center" border="0" cellpadding="0" cellspacing="0"> <tr> <td nowrap="">Dim<sub>{<em>G<sub>n</sub>}</em></sub></td></tr></table></p>(<em>H</em>) = <td align="center" nowrap=""><sup>lim inf</sup><sub style="position: relative; left: -2.3em;"><sub><em>n&rightarrow;∞</em></sub></sub></td> <td align="center" nowrap="">log<sub><em>p</em></sub> | <em>HG</em><sup>(<em>k</em>)</sup><sub style="position: relative; left: -.8em;"><em>n</em></sub>/ <em>G</em><sup>(<em>k</em>)</sup><sub style="position: relative; left: -.8em;"><em>n</em></sub>|<hr noshade="" size="1">log<sub><em>p</em></sub> | (<em>G</em> / <em>G</em><sub><em>n</em></sub>)<sup>(<em>k</em>)</sup> |</hr></td> <td nowrap=""> ·</td> <p>In this setting, a rather natural way to define the metric is by using the filtration <em>G<sub>n</sub></em> = <sup>/</sup><sub style="position: relative; left: -.3em;">\</sub><em>x<sup>p<sup>n</sup></sup></em> : <em>x</em> &in; <em>G</em></p><sup>\</sup><sub style="position: relative; left: -.28em;">/</sub>. For this filtration, we ask whether for any word <em>w</em> in <em>k</em> variables, Dim<sub>{<em>G<sub>n</sub></em>}</sub> <em>S<sub>w</sub></em>(1) ≥ <em>k</em> - 1/<em>k</em>. We show that for free pro-<em>p</em> groups, using the filtration given by its dimension subgroups, this is not true in general.
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