Let be a group equipped with a homogeneous norm ( is necessarily abelian and torsion-free). Then this norm extends to a norm on the -vectorspace . In detail, every element of has a representative of the form with and we set .

Now let $\overline{A_\mathbb{Q}}$ be the metric completion of . This is still a normed group into which embeds, and is in fact an -vectorspace: if and are Cauchy sequences and then so is and it is easy to check that its equivalence class depends only on the equivalence classes on the original sequences. Since also it also follows that the norm is -homogenous. In summary, is naturally a complete normed -vector space, that is a Banach space.

Finally, there is a natural map of -vectorspaces . This map is injective since our extension of the norm was still a norm. Since the RHS is compatibly an -vectorspace, this induces a further map of -vectorspaces

.

However, the latter map need not be injective! In other words, when we pull back the norm from to , the result need only be a seminorm.

For example, if we start with the norm on then the same formula defines a norm on but only a seminorm on . To get a norm we need to divide by the subspace which is compatible with everything we’ve said since this subspace is disjoint from the image of here so and still inject in the quotient (isometrically, as they must).

]]>arbitrary elements). ]]>

,

assuming that satisfies homogeneity and the triangle inequality on the nose. Thus .

Doing a more refined analysis along the lines of Proposition 1 in the second blog post could be interesting, but I haven’t done this yet and find it hard to say whether it would have the potential to lead to an alternative proof or not.

]]>this feels rather late in the day (or sometime the next day) but

still: why can’t we argue thus….

Given with homogeneous (pseudo-) length function ,

let be your favourite upper bound that holds for all

commutators, thus for all . By

Culler’s identity

(sometimes seen in introductions to stable commutator length)

we have , thus we now know holds for any

commutator. So now use the argument recursively to get

for all and hence .

There were two limitations of the way the computer proof was done:

* While the use of conjugacy invariance and triangle inequality was optimal and algorithmic, of which elements to take powers was manually specified by me. This should have been made smart, and would have soon enough except the extreme smartness of the people in this polymath group made this redundant (problem was solved within 24 hours of the first posted computer proof).

* More importantly, I used [domain specific foundations](https://github.com/siddhartha-gadgil/Superficial/blob/master/src/main/scala/freegroups/LinNormBound.scala), which could encode only one kind of proof, that a specific word has length bounded by a specific number. This rules out in particular both firmulas for bounds that are quatified (and so must invlove variables) and recursion/induction. To show that such results can be at least _encoded_ I formalized the [internal repetition trick](http://siddhartha-gadgil.github.io/ProvingGround/tuts/LengthFunctions.html).

More generally, where a computer helped was in following instructions of the form “try these method in lots of cases in lots of ways and give me the best proof for thess cases (or where we got a strong result)”. It is obvious that the “lots of cases” and “lots of ways” are much bigger numbers for computers than by hand. The question is how general one can be with “these methods”. I do think even in practice a lot of methods can be encoded, and this is underutilized as people underestimate this. In principle, in the era of Homotopy type theory and Deep learning, presumably every method can be encoded.

]]>Is there any possibility to summarize in more details of computer-assistance (by any of the participants; and on a more abstract level maybe, i am not an expert in this field)? Also maybe in the other polymath projects? Is there a structure of mathematical problems that are quite good for computer-assisted proofs? What where the insights, has it “just” been the calculation of difficult formulars – i.e. standard Mathematica problems? This is super interesting!

]]>Mainly as an exercise for my benefit. I have formalized (in the sense of computer verified, but with idiosyncratic foundations) the internal repetition lemma at http://siddhartha-gadgil.github.io/ProvingGround/tuts/LengthFunctions.html ]]>