Low-dimensional linear representations of 𝐴𝑢𝑡𝐹_{𝑛},𝑛≥3

Abstract

Abstract. We classify all complex representations of Aut Fn, the automorphism group of the free group Fn (n ≥ 3), of dimension ≤ 2n − 2. Among those representations is a new representation of dimension n + 1 which does not vanish on the group of inner automorphisms.

The rhombic dodecahedron and semisimple actions of Aut(Fn) on CAT(0) spaces

“…Aut(F n ) and Out(F n ) are not Kähler groups One can deduce from Corollary 8.3 that the standard representation Aut(F n ) → GL(n, Z) (given by the action of Aut(F n ) on H 1 (F n , Z)) cannot be deformed locally to anything but a conjugate representation. A stonger result was proved by Dyer and Formanek [10] (see also [16] Theorem 1.2): every representation Aut(F n ) → GL(n, C) factors through the standard representation. Using Proposition 3.1 of [8] one can extend the proof in [16] to cover the unique index-2 subgroup SAut(F n ) ⊂ Aut(F n ).…”

“…A stonger result was proved by Dyer and Formanek [10] (see also [16] Theorem 1.2): every representation Aut(F n ) → GL(n, C) factors through the standard representation. Using Proposition 3.1 of [8] one can extend the proof in [16] to cover the unique index-2 subgroup SAut(F n ) ⊂ Aut(F n ).…”

“…Actions of the second type arise from linear representations Aut(F n ) → GL(d, R) via the action of GL(d, R) on its symmetric space. The fact that Nielsen transformations must act as neutral parabolics imposes constraints on the representation theory of Aut(F n ) that supplement [10] and [16]. In Section 9 we explain how the rigidity of the standard linear representation [10] can be combined with Simpson's results [17] to prove: …”

Abelian covers of graphs and maps between outer automorphism groups of free groups

“…In the case where n is odd, our proof of Theorem C does not imply an analogue of Theorem 7.9 because the proof of Proposition 7.8 relies heavily on the ambient structure of Out(F n ) and in particular on its low dimensional representation theory. That proof begs the question of whether a closer study of the representation theory of Out(F n ), extending Theorem 3.1 of [20] and paying particular attention to the −1 eigenspaces of the ε i and ∆, might allow one to improve the bound m ≤ 2n in Theorem C without having to classify all homomorphisms W n , G n → Out(F m ) in the expanded range. This idea is pursued by Dawid Kielak in his Oxford doctoral thesis, cf.…”

Outer automorphism groups of free groups: linear and free representations

“…To put our theorems in context, let us mention the work of Potapchik and Rapinchuk [17]. They study complex linear representations of Aut(F n ) in dimension at most 2n−2.…”