[math-fun] Some serious algebraic weirdness for higher integers
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.
On Sat, Jan 12, 2019 at 9:34 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.
My favorite fact about free groups is that these are the *only* subgroups of F_n; that is, any subgroup of a free group is free. If you think of this as a fact about algebra, I don't see how you'd go about proving it. But if you think of it as a fact about topology, it has a very simple explanation: The fundamental group of a graph is a free group. A covering space of a graph is a graph. QED. Andy
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.
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participants (3)
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Andy Latto -
Dan Asimov -
Paul Palmer