The paper is "Integral Euler characteristic of Out(F_11)" by Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki, available at <https://arxiv.org/pdf/1405.4063.pdf>. —Dan Paul Palmer wrote: ----- More details on the Morita paper? I couldn't find it. Thanks On Sat, Jan 12, 2019, 8:34 PM Dan Asimov <dasimov@earthlink.net> wrote:
Just something amusing I'm learning about — For each integer let F_n be the free group on n generators, nonabelian for n >= 2.°
These groups are utterly natural yet quite intriguing. Like the fact that F_2 contains an isomorphic copy of F_n for any n, even if n = aleph_0.
So let's ask about its *automorphism* group, the group of all self-homomorphisms a : F_n —> F_n that are invertible and have inverse that is also a homomorphism. (Too much in a rush at the moment to check if the last condition is redundant.)
Anyhow, all groups act on themselves by inner automorphisms. For any g in a group g there is the inner automorphism x —> g x g^(-1) of G for any x in G.
Definition: ----------- Let Inn(G) be the group of all inner automorphisms of G. (It's sometimes isomorphic to G, other times not.)
So, the interesting automorphism thing is the *outer* automorphisms, the quotient group
Out(G) = Aut(G) / Inn(G)
Anyhow, people have compared Out(F_n) for n = 2, 3, 4, .... One way to compare them is to consider the INTEGER X(Out(F_n)) where X(G), for any group G, denotes the Euler characteristic of the topological space K(G,1). (The space is uniquely defined only up to homotopy type, but that's enough to uniquely determine its Euler characteristic.
Anyhoo, the point of the above is to convince you that the table below is a natural thing to consider:
n 2 3 4 5 6 7 8 9 10 11 ————————————————————————————————————————————————————————————————————————— X(Out(F_n)) 1 1 2 1 2 1 1 -24 -121 -1202
(From a 2014 paper by Shigeyuki Morita et al.) This is seriously weird.
—Dan —————————————————————————————————————————————————————————————————————————— * If you don't know about the free group F_n: This is the smallest group having n distinct generators and no relations (which defines it up to isomorphism. F_1 = Z but for n >= 2 they're nonabelian. F_n if made abelian becomes the direct sum of Z n times.