[Paper Review] The linearity problem for the unitriangular automorphism groups of free groups

В. А. Романьков|arXiv (Cornell University)|Sep 11, 2013

Advanced Differential Equations and Dynamical Systems16 references3 citations

TL;DR

This paper resolves the linearity problem for unitriangular automorphism groups of free groups by proving that such groups admit a faithful matrix representation over a field if and only if the free group has rank at most 3. The authors use HNN extensions and the Formanek–Procesi method to show that $ U_4 $ is not linear, and by extension, $ U_n $ is not linear for $ n \geq 4 $, completing a classification of linearity across all relatively free groups.

ABSTRACT

We prove that the unitriangular automorphism group of a free group of rank $n$ has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if $n \leq 3.$ Thus, we have completed a description of relatively free groups with linear the unitriangular automorphism groups. This description was initiated by Erofeev and the author in \cite{Erofeev}, where proper varieties of groups have been considered.

Motivation & Objective

• To complete the classification of varieties of groups for which the unitriangular automorphism group is linear.
• To resolve the remaining open case in the linearity problem for unitriangular automorphism groups, specifically when the variety is that of all groups.
• To determine whether $ U_n $, the unitriangular automorphism group of the free group $ F_n $, is linear for $ n \geq 4 $.

Proposed method

• Use the HNN extension construction $ \mathcal{H}(G) = \langle G \times G, t \mid t(g,g)t^{-1} = (1,g) \rangle $ to analyze linearity obstructions.
• Apply Theorem 3.2 of Brendle and Hamidi-Tehrani to show that if a group maps onto a quotient $ \mathcal{H}(F_2)/N $ with non-nilpotent-by-abelian-by-finite image of $ F_2 \times \{1\} $, then it is not linear.
• Construct a subgroup $ H \leq U_4 $ isomorphic to a quotient of $ \mathcal{H}(F_2) $, using generators $ \lambda_{3,1}, \lambda_{3,2}, \lambda_{4,1}, \lambda_{4,2}, \lambda_{4,3} $, and verify the required relations.
• Show that $ H \cong \mathcal{H}(F_2)/N $ with the image of $ F_2 \times \{1\} $ not nilpotent-by-abelian-by-finite, thus proving $ U_4 $ is not linear.
• Establish that $ U_4' $, the derived subgroup of $ U_4 $, is isomorphic to $ W_4 = \langle \lambda_{3,j}, \lambda_{4,l} \mid j=1,2; l=1,2,3 \rangle $, and hence $ U_4' $ is not linear.
• Generalize the result to $ n \geq 4 $ by showing that $ \gamma_{n-2}(U_n) $, the $ (n-2) $-th term of the lower central series, is not linear.

Experimental results

Research questions

• RQ1 For which varieties of groups is the unitriangular automorphism group $ U_n $ linear?
• RQ2 Is the unitriangular automorphism group $ U_n $ of the free group $ F_n $ linear for $ n \geq 4 $?
• RQ3 Does the linearity of $ U_3 $ imply the linearity of $ U_n $ for all $ n \geq 4 $?
• RQ4 What is the structure of the derived subgroup $ U_n' $, and is it linear for $ n \geq 4 $?
• RQ5 Can the non-linearity of $ U_n $ be detected via HNN extension quotients?

Key findings

• The unitriangular automorphism group $ U_n $ of the free group $ F_n $ is linear if and only if $ n \leq 3 $, completing the classification of linearity for all relatively free groups.
• The derived subgroup $ U_4' $ is not linear, and it is isomorphic to the subgroup $ W_4 = \langle \lambda_{3,1}, \lambda_{3,2}, \lambda_{4,1}, \lambda_{4,2}, \lambda_{4,3} \rangle $, which is a quotient of $ \mathcal{H}(F_2) $ with non-nilpotent-by-abelian-by-finite image.
• For every $ n \geq 4 $, the $ (n-2) $-th term $ \gamma_{n-2}(U_n) $ of the lower central series of $ U_n $ is not linear.
• The subgroup $ U_4' $ is also the stabilizer of $ f_1 $ in $ U_4 $, showing that this normal subgroup is not linear.
• The result implies that $ U_3 $ being linear does not imply $ U_n $ is linear for $ n \geq 4 $, contradicting a potential generalization from the $ U_3 $ case.
• The paper conjectures the existence of varieties $ \mathcal{G}_m $ such that $ U_n $ is linear if and only if $ n \leq m $, with $ m \geq 3 $.