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The Polymath14 online collaboration has uploaded to the arXiv its paper “Homogeneous length functions on groups“, submitted to Algebra & Number Theory. The paper completely classifies homogeneous length functions {\| \|: G \rightarrow {\bf R}^+} on an arbitrary group {G = (G,\cdot,e,()^{-1})}, that is to say non-negative functions that obey the symmetry condition {\|x^{-1}\| = \|x\|}, the non-degeneracy condition {\|x\|=0 \iff x=e}, the triangle inequality {\|xy\| \leq \|x\| + \|y\|}, and the homogeneity condition {\|x^2\| = 2\|x\|}. It turns out that these norms can only arise from pulling back the norm of a Banach space by an isometric embedding of the group. Among other things, this shows that {G} can only support a homogeneous length function if and only if it is abelian and torsion free, thus giving a metric description of this property.

The proof is based on repeated use of the homogeneous length function axioms, combined with elementary identities of commutators, to obtain increasingly good bounds on quantities such as {\|[x,y]\|}, until one can show that such norms have to vanish. See the previous post for a full proof. The result is robust in that it allows for some loss in the triangle inequality and homogeneity condition, allowing for some new results on “quasinorms” on groups that relate to quasihomomorphisms.

As there are now a large number of comments on the previous post on this project, this post will also serve as the new thread for any final discussion of this project as it winds down.

In the tradition of “Polymath projects“, the problem posed in the previous two blog posts has now been solved, thanks to the cumulative effect of many small contributions by many participants (including, but not limited to, Sean Eberhard, Tobias Fritz, Siddharta Gadgil, Tobias Hartnick, Chris Jerdonek, Apoorva Khare, Antonio Machiavelo, Pace Nielsen, Andy Putman, Will Sawin, Alexander Shamov, Lior Silberman, and David Speyer). In this post I’ll write down a streamlined resolution, eliding a number of important but ultimately removable partial steps and insights made by the above contributors en route to the solution.

Theorem 1 Let {G = (G,\cdot)} be a group. Suppose one has a “seminorm” function {\| \|: G \rightarrow [0,+\infty)} which obeys the triangle inequality

\displaystyle \|xy \| \leq \|x\| + \|y\|

for all {x,y \in G}, with equality whenever {x=y}. Then the seminorm factors through the abelianisation map {G \mapsto G/[G,G]}.

Proof: By the triangle inequality, it suffices to show that {\| [x,y]\| = 0} for all {x,y \in G}, where {[x,y] := xyx^{-1}y^{-1}} is the commutator.

We first establish some basic facts. Firstly, by hypothesis we have {\|x^2\| = 2 \|x\|}, and hence {\|x^n \| = n \|x\|} whenever {n} is a power of two. On the other hand, by the triangle inequality we have {\|x^n \| \leq n\|x\|} for all positive {n}, and hence by the triangle inequality again we also have the matching lower bound, thus

\displaystyle \|x^n \| = n \|x\|

for all {n > 0}. The claim is also true for {n=0} (apply the preceding bound with {x=1} and {n=2}). By replacing {\|x\|} with {\max(\|x\|, \|x^{-1}\|)} if necessary we may now also assume without loss of generality that {\|x^{-1} \| = \|x\|}, thus

\displaystyle \|x^n \| = |n| \|x\| \ \ \ \ \ (1)

 

for all integers {n}.

Next, for any {x,y \in G}, and any natural number {n}, we have

\displaystyle \|yxy^{-1} \| = \frac{1}{n} \| (yxy^{-1})^n \|

\displaystyle = \frac{1}{n} \| y x^n y^{-1} \|

\displaystyle \leq \frac{1}{n} ( \|y\| + n \|x\| + \|y\|^{-1} )

so on taking limits as {n \rightarrow \infty} we have {\|yxy^{-1} \| \leq \|x\|}. Replacing {x,y} by {yxy^{-1},y^{-1}} gives the matching lower bound, thus we have the conjugation invariance

\displaystyle \|yxy^{-1} \| = \|x\|. \ \ \ \ \ (2)

 

Next, we observe that if {x,y,z,w} are such that {x} is conjugate to both {wy} and {zw^{-1}}, then one has the inequality

\displaystyle \|x\| \leq \frac{1}{2} ( \|y \| + \| z \| ). \ \ \ \ \ (3)

 

Indeed, if we write {x = swys^{-1} = t zw^{-1} t^{-1}} for some {s,t \in G}, then for any natural number {n} one has

\displaystyle \|x\| = \frac{1}{2n} \| x^n x^n \|

\displaystyle = \frac{1}{2n} \| swy \dots wy s^{-1}t zw^{-1} \dots zw^{-1} t^{-1} \|

where the {wy} and {zw^{-1}} terms each appear {n} times. From (2) we see that conjugation by {w} does not affect the norm. Using this and the triangle inequality several times, we conclude that

\displaystyle \|x\| \leq \frac{1}{2n} ( \|s\| + n \|y\| + \| s^{-1} t\| + n \|z\| + \|t^{-1} \| ),

and the claim (3) follows by sending {n \rightarrow \infty}.

The following special case of (3) will be of particular interest. Let {x,y \in G}, and for any integers {m,k}, define the quantity

\displaystyle f(m,k) := \| x^m [x,y]^k \|.

Observe that {x^m [x,y]^k} is conjugate to both {x (x^{m-1} [x,y]^k)} and to {(y^{-1} x^m [x,y]^{k-1} xy) x^{-1}}, hence by (3) one has

\displaystyle \| x^m [x,y]^k \| \leq \frac{1}{2} ( \| x^{m-1} [x,y]^k \| + \| y^{-1} x^{m} [x,y]^{k-1} xy \|)

which by (2) leads to the recursive inequality

\displaystyle f(m,k) \leq \frac{1}{2} (f(m-1,k) + f(m+1,k-1)).

We can write this in probabilistic notation as

\displaystyle f(m,k) \leq {\bf E} f( (m,k) + X )

where {X} is a random vector that takes the values {(-1,0)} and {(1,-1)} with probability {1/2} each. Iterating this, we conclude in particular that for any large natural number {n}, one has

\displaystyle f(0,n) \leq {\bf E} f( Z )

where {Z := (0,n) + X_1 + \dots + X_{2n}} and {X_1,\dots,X_{2n}} are iid copies of {X}. We can write {Z = (1,-1/2) (Y_1 + \dots + Y_{2n})} where Y_1,\dots,Y_{2n} = \pm 1 are iid signs.  By the triangle inequality, we thus have

\displaystyle f( Z ) \leq |Y_1+\dots+Y_{2n}| (\|x\| + \frac{1}{2} \| [x,y] \|),

noting that Y_1+\dots+Y_{2n} is an even integer.  On the other hand, Y_1+\dots+Y_{2n} has mean zero and variance 2n, hence by Cauchy-Schwarz

\displaystyle f(0,n) \leq \sqrt{2n}( \|x\| + \frac{1}{2} \| [x,y] \|).

But by (1), the left-hand side is equal to {n \| [x,y]\|}. Dividing by {n} and then sending {n \rightarrow \infty}, we obtain the claim. \Box

The above theorem reduces such seminorms to abelian groups. It is easy to see from (1) that any torsion element of such groups has zero seminorm, so we can in fact restrict to torsion-free groups, which we now write using additive notation {G = (G,+)}, thus for instance {\| nx \| = |n| \|x\|} for {n \in {\bf Z}}. We think of {G} as a {{\bf Z}}-module. One can then extend the seminorm to the associated {{\bf Q}}-vector space {G \otimes_{\bf Z} {\bf Q}} by the formula {\|\frac{a}{b} x\| := \frac{a}{b} \|x\|}, and then to the associated {{\bf R}}-vector space {G \otimes_{\bf Z} {\bf R}} by continuity, at which point it becomes a genuine seminorm (provided we have ensured the symmetry condition {\|x\| = \|x^{-1}\|}). Conversely, any seminorm on {G \otimes_{\bf Z} {\bf R}} induces a seminorm on {G}. (These arguments also appear in this paper of Khare and Rajaratnam.)

 

This post is a continuation of the previous post, which has attracted a large number of comments. I’m recording here some calculations that arose from those comments (particularly those of Pace Nielsen, Lior Silberman, Tobias Fritz, and Apoorva Khare). Please feel free to either continue these calculations or to discuss other approaches to the problem, such as those mentioned in the remaining comments to the previous post.

Let {F_2} be the free group on two generators {a,b}, and let {\| \|: F_2 \rightarrow {\bf R}^+} be a quantity obeying the triangle inequality

\displaystyle \| xy\| \leq \|x \| + \|y\|

and the linear growth property

\displaystyle \| x^n \| = |n| \| x\|

for all {x,y \in F_2} and integers {n \in {\bf Z}}; this implies the conjugation invariance

\displaystyle \| y^{-1} x y \| = \|x\|

or equivalently

\displaystyle \| xy \| = \| yx\|

We consider inequalities of the form

\displaystyle \| xyx^{-1}y^{-1} \| \leq \alpha \|x\| + \beta \| y\| \ \ \ \ \ (1)

or

\displaystyle \| xyx^{-2}y^{-1} \| \leq \gamma \|x\| + \delta \| y\| \ \ \ \ \ (2)

for various real numbers {\alpha,\beta,\gamma,\delta}. For instance, since

\displaystyle \| xyx^{-1}y^{-1} \| \leq \| xyx^{-1}\| + \|y^{-1} \| = \|y\| + \|y\|

we have (1) for {(\alpha,\beta) = (2,0)}. We also have the following further relations:

Proposition 1

  • (i) If (1) holds for {(\alpha,\beta)}, then it holds for {(\beta,\alpha)}.
  • (ii) If (1) holds for {(\alpha,\beta)}, then (2) holds for {(\alpha+1, \frac{\beta}{2})}.
  • (iii) If (2) holds for {(\gamma,\delta)}, then (1) holds for {(\frac{2\gamma}{3}, \frac{2\delta}{3})}.
  • (iv) If (1) holds for {(\alpha,\beta)} and (2) holds for {(\gamma,\delta)}, then (1) holds for {(\frac{2\alpha+1+\gamma}{4}, \frac{\delta+\beta}{4})}.

Proof: For (i) we simply observe that

\displaystyle \| xyx^{-1} y^{-1} \| = \| (xyx^{-1} y^{-1})^{-1} \| = \| y^{-1} x^{-1} y x \| = \| y x y^{-1} x^{-1} \|.

For (ii), we calculate

\displaystyle \| xyx^{-2}y^{-1} \| = \frac{1}{2}\| (xyx^{-2}y^{-1})^2 \|

\displaystyle = \frac{1}{2} \| (xyx^{-2}y^{-1} x) (yx^{-2} y^{-1}) \|

\displaystyle \leq \frac{1}{2} (\| xyx^{-2}y^{-1} x\| + \|yx^{-2} y^{-1}\|)

\displaystyle \leq \frac{1}{2} ( \| x^2 y x^{-2} y^{-1} \| + 2 \|x\| )

\displaystyle \leq \frac{1}{2} ( 2 \alpha \|x\| + \beta \|y\| + 2 \|x\|)

giving the claim.

For (iii), we calculate

\displaystyle \| xyx^{-1}y^{-1}\| = \frac{1}{3} \| (xyx^{-1}y^{-1})^3 \|

\displaystyle = \frac{1}{3} \| (xyx) (x^{-2} y^{-1} xy) (xyx)^{-1} (x^2 y x^{-1} y^{-1}) \|

\displaystyle \leq \frac{1}{3} ( \| x^{-2} y^{-1} xy\| + \| x^2 y x^{-1} y^{-1}\| )

\displaystyle = \frac{1}{3} ( \| xy x^{-2} y^{-1} \| + \|x^{-1} y^{-1} x^2 y \| )

\displaystyle \leq \frac{1}{3} ( \gamma \|x\| + \delta \|y\| + \gamma \|x\| + \delta \|y\|)

giving the claim.

For (iv), we calculate

\displaystyle \| xyx^{-1}y^{-1}\| = \frac{1}{4} \| (xyx^{-1}y^{-1})^4 \|

\displaystyle = \frac{1}{4} \| (xy) (x^{-1} y^{-1} x) (y x^{-1} y^{-1}) (xyx^{-1}) (xy)^{-1} (x^2yx^{-1}y^{-1}) \|

\displaystyle \leq \frac{1}{4} ( \| (x^{-1} y^{-1} x) (y x^{-1} y^{-1}) (xyx^{-1}) \| + \|x^2yx^{-1}y^{-1}\| )

\displaystyle \leq \frac{1}{4} ( \|(y x^{-1} y^{-1}) (xy^{-1}x^{-1})(x^{-1} y x) \| + \gamma \|x\| + \delta \|y\|)

\displaystyle \leq \frac{1}{4} ( \|x\| + \|(xy^{-1}x^{-1})(x^{-1} y x) \| + \gamma \|x\| + \delta \|y\|)

\displaystyle = \frac{1}{4} ( \|x\| + \|x^{-2} y x^2 y^{-1} \|+ \gamma \|x\| + \delta \|y\|)

\displaystyle \leq \frac{1}{4} ( \|x\| + 2\alpha \|x\| + \beta \|y\| + \gamma \|x\| + \delta \|y\|)

giving the claim. \Box

Here is a typical application of the above estimates. If (1) holds for {(\alpha,\beta)}, then by part (i) it holds for {(\beta,\alpha)}, then by (ii) (2) holds for {(\beta+1,\frac{\alpha}{2})}, then by (iv) (1) holds for {(\frac{3\beta+2}{4}, \frac{3\alpha}{8})}. The map {(\alpha,\beta) \mapsto (\frac{3\beta+2}{4}, \frac{3\alpha}{8})} has fixed point {(\alpha,\beta) = (\frac{16}{23}, \frac{6}{23})}, thus

\displaystyle \| xyx^{-1}y^{-1} \| \leq \frac{16}{23} \|x\| + \frac{6}{23} \|y\|.

For instance, if {\|a\|, \|b\| \leq 1}, then {\|aba^{-1}b^{-1} \| \leq 22/23 = 0.95652\dots}.

Here is a curious question posed to me by Apoorva Khare that I do not know the answer to. Let {F_2} be the free group on two generators {a,b}. Does there exist a metric {d} on this group which is

  • bi-invariant, thus {d(xg,yg)=d(gx,gy) = d(x,y)} for all {x,y,g \in F_2}; and
  • linear growth in the sense that {d(x^n,1) = n d(x,1)} for all {x \in F_2} and all natural numbers {n}?

By defining the “norm” of an element {x \in F_2} to be {\| x\| := d(x,1)}, an equivalent formulation of the problem asks if there exists a non-negative norm function {\| \|: F_2 \rightarrow {\bf R}^+} that obeys the conjugation invariance

\displaystyle  \| gxg^{-1} \| = \|x \| \ \ \ \ \ (1)

for all {x,g \in F_2}, the triangle inequality

\displaystyle  \| xy \| \leq \| x\| + \| y\| \ \ \ \ \ (2)

for all {x,y \in F_2}, and the linear growth

\displaystyle  \| x^n \| = |n| \|x\| \ \ \ \ \ (3)

for all {x \in F_2} and {n \in {\bf Z}}, and such that {\|x\| > 0} for all non-identity {x \in F_2}. Indeed, if such a norm exists then one can just take {d(x,y) := \| x y^{-1} \|} to give the desired metric.

One can normalise the norm of the generators to be at most {1}, thus

\displaystyle  \| a \|, \| b \| \leq 1.

This can then be used to upper bound the norm of other words in {F_2}. For instance, from (1), (3) one has

\displaystyle  \| aba^{-1} \|, \| b^{-1} a b \|, \| a^{-1} b^{-1} a \|, \| bab^{-1}\| \leq 1.

A bit less trivially, from (3), (2), (1) one can bound commutators as

\displaystyle  \| aba^{-1} b^{-1} \| = \frac{1}{3} \| (aba^{-1} b^{-1})^3 \|

\displaystyle  = \frac{1}{3} \| (aba^{-1}) (b^{-1} ab) (a^{-1} b^{-1} a) (b ab^{-1}) \|

\displaystyle  \leq \frac{4}{3}.

In a similar spirit one has

\displaystyle  \| aba^{-2} b^{-1} \| = \frac{1}{2} \| (aba^{-2} b^{-1})^2 \|

\displaystyle  = \frac{1}{2} \| (aba^{-1}) (a^{-1} b^{-1} a) (ba^{-1} b^{-1}) (ba^{-1} b^{-1}) \|

\displaystyle  \leq 2.

What is not clear to me is if one can keep arguing like this to continually improve the upper bounds on the norm {\| g\|} of a given non-trivial group element {g} to the point where this norm must in fact vanish, which would demonstrate that no metric with the above properties on {F_2} would exist (and in fact would impose strong constraints on similar metrics existing on other groups as well). It is also tempting to use some ideas from geometric group theory (e.g. asymptotic cones) to try to understand these metrics further, though I wasn’t able to get very far with this approach. Anyway, this feels like a problem that might be somewhat receptive to a more crowdsourced attack, so I am posing it here in case any readers wish to try to make progress on it.

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