Notes on the Dehn Function of Π Aut(F3)

Rylee Lyman January 27, 2019

Bridson and Vogtmann in [BV95] show that Aut(F3) and Out(F3) each have exponential n −n n −1 −n −1 Dehn function. Their proof relies on the words wn = t at bt a t b and an argument of Gersten [Ger92a] [Ger92b]. We ﬁx Fn = hx1, . . . , xn | i. Bridson–Vogtmann’s words are    x 7→ x x 7→ x x 7→ x2x  1 1  1 1  1 1 2 a = x2 7→ x2 b = x2 7→ x2 t = x2 7→ x1x2    x3 7→ x1x3 x3 7→ x3x2 x3 7→ x3

n −n n We see that a and b commute—in fact, t at maps x3 to t (a)x3 while ﬁxing all the other generators, so it also commutes with b. The key observation is that the image of ha, b, ti 2 under the projection Aut(F3) → GL3(Z) is isomorphic to the mapping torus Z oϕ Z of a + hyperbolic element of SL2(Z), so Remark 5.10 of [Ger92b], together with 10.4 of [ECH 92] shows that the area of a van Kampen diagram for wn grows exponentially in n. We are interesting in contrasting this result to the situation for EΠ Aut(F3), the subgroup of elementary palindromic automorphisms—i.e. those sending each generator to a palindrome spelled the same forwards and backwards. Since this subgroup surjects onto the principal level 2 congruence subgroup of SL3(Z) (see for instance [FT18], although the proof of this fact is elementary), candidate hyperbolic elements of SL2(Z) to play the role of t in Bridson–Vogtmann’s proof abound. Perhaps the simplest is  x1 7→ x1x2x1x2x1 0  t = x2 7→ x1x2x1  x3 7→ x3

However, since a and b are not palindromic, we’ll need to ﬁnd candidates to replace them as well. A ﬁrst try might yield   x1 7→ x1 x1 7→ x1 0  0  a = x2 7→ x2 b = x2 7→ x2   x3 7→ x1x3x1 x3 7→ x2x3x2

0 which certainly gives the correct image in GL3(Z). The problem, however, is that a and 0 0n 0 0−n 0 0n 0n b no longer commute. In fact t a t b takes x3 to x2t (x1)x3t (x1)x2 and ﬁxes x1 and 0 0n 0 0−n 0n 0n x2, while b t a t still ﬁxes x1 and x2, but takes x3 to t (x1)x2x3x2t (x1). 2 It appears that there is not enough room in Π Aut(F3) to encode both Z and a hyperbolic element of SL2(Z), or, properly speaking, an element that will project to one.

1 We can amend the situation somewhat by going up a rank to Π Aut(F4). Let us deﬁne    x1 7→ x1 x1 7→ x1 x1 7→ x1x2x1x2x1    x 7→ x x 7→ x x 7→ x x x a00 = 2 2 b00 = 2 2 t00 = 2 1 2 1 x3 7→ x1x3x1 x3 7→ x3 x3 7→ x3    x4 7→ x4 x4 7→ x2x4x2 x4 7→ x4

00n 00 00−n 00n 00−n Now, t a t sends x3 to t (x1)x3t (x1) and ﬁxes the other three generators, so t00na00t00−n and b00 commute. But sadly, we’ve exchanged one diﬃculty for another: although Gersten’s argument gives an exponential lower bound on the area of a van Kampen diagram for the image in Z2 o Z 00n 00 00−n 00 of [t a t , b ], because the Dehn function for GL4(Z) remains unknown, we cannot guarantee that this area is optimal.

References

[BV95] Martin R. Bridson and Karen Vogtmann. On the geometry of the automorphism group of a free group. Bull. London Math. Soc., 27(6):544–552, 1995. [ECH+92] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. [FT18] Neil Fullarton and Anne Thomas. Palindromic automorphisms of right-angled Artin groups. Groups Geom. Dyn., 12(3):865–887, 2018. [Ger92a] S. M. Gersten. Bounded cocycles and combings of groups. Internat. J. Algebra Comput., 2(3):307–326, 1992.

[Ger92b] S. M. Gersten. Dehn functions and l1-norms of ﬁnite presentations. In Algorithms and classiﬁcation in combinatorial group theory (Berkeley, CA, 1989), volume 23 of Math. Sci. Res. Inst. Publ., pages 195–224. Springer, New York, 1992.