Proposition 1.

Take any natural numbers n≤m. Let p1 be a finitely generated metric on Fn and let p2 be a finitely generated metric on Fm such that p1≥p2↾Fn and p1 is δ-close to p2↾Fn, for some δ>0. Then there exists a finitely generated metric p on Fn′∗Fm, with free generators {f1′,…,fn′,f1,…,fn,…,fm} such that p↾Fn′≅p1↾Fn, p↾Fm≅p2↾Fm and for every i≤n we have p⁢(fi′,fi)≤δ.

Remark 2.

We note that in the original statement there was an equality p⁢(fi′,fi)=δ. While it is possible to prove the equality (under the assumption stated just before the statement of Proposition 2.15 in [1] that δ≤p1⁢(fi,1)+p2⁢(fi,1)), in order to keep the size of the corrigendum appropriate we note that the inequality p⁢(fi′,fi)≤δ is sufficient for Embedding Construction 2.3 in [1]. Indeed, the reader can readily verify that the construction employs a sequence (εi)i with ∑iεi<∞, and sequences of group generators (fi(n))n satisfying d⁢(fi(n),fi(n+1))=εn in order to be Cauchy. Obviously, if we ensure d⁢(fi(n),fi(n+1))≤εn, the sequence is Cauchy too.

Definition 3.

If G is a group, then a function λ:G→[0,∞) is a called a conjugation-invariant norm (or shortly a CI-norm) if

1. λ⁢(g)=0 if and only if g=1G,

2. λ⁢(g)=λ⁢(g-1) for every g∈G,

3. λ⁢(g⋅h)≤λ⁢(g)+λ⁢(h) for every g,h∈G,

4. λ⁢(h-1⋅g⋅h)=λ⁢(g) for every g,h∈G.

It is clear that if d is a bi-invariant metric on G, then λd⁢(g)=d⁢(g,1) is a conjugation-invariant norm on G, and conversely, if λ is a conjugation-invariant norm on G, then dλ⁢(g,h)=λ⁢(g-1⋅h) (=λ⁢(h⋅g-1)) is a bi-invariant metric on G.

Definition 4.

Let A⊆G be a symmetric generating subset of a group G. We say that a conjugation-invariant norm λ on G is generated by values on A⊆G if for every g∈G we have

where gh, for g,h∈G, denotes the element h-1⋅g⋅h.

If moreover A is finite, we say that λ is finitely generated, and the infimum in the above definition can be then replaced by the minimum.

Observation 5.

It is straightforward to check that if d is a bi-invariant metric on G generated by values on some subset A⊆G2, then the corresponding CI-norm λd is generated by values on {g-1⁢h,h-1⁢g:(g,h)∈A}. Conversely, if λ is a CI-norm generated by values on some A⊆G, then dλ is generated by values on the set A′={(a,1G),(1G,a):a∈A}.

Definition 6 (Match).

Let w=w1⁢…⁢wn be a word in the alphabet consisting of the free generators of G and their inverses. A match for w is a bijection ρ on {1,…,n} satisfying the following properties. We note that in order to simplify the notation, we shall often freely identify ρ as a bijection on {1,…,n} with a bijection on {w1,…,wn}, i.e. write ρ⁢(wi)=wj instead of wρ⁢(i)=wj.

1. for every i≤n, we have ρ⁢(ρ⁢(i))=i, i.e. ρ∘ρ=id,

2. for no i,j≤n, we have i<j<ρ⁢(i)<ρ⁢(j),

3. if for some i≤n, ρ⁢(wi)≠wi, then ρ⁢(wi)=wi-1 (the letter ρ⁢(wi) is the inverse of the letter wi).

Remark 7.

The interpretation of this definition of a match ρ is that ρ describes how the word w is built from its subwords using the operations of concatenation and conjugation. For example if w=a⁢b⁢a-1⁢c and a match ρ is defined as ρ⁢(a)=a-1, ρ⁢(b)=b, ρ⁢(c)=c, then ρ provides the information that w was built by conjugating the letter b by the letter a (or a-1) and then concatenating with c.

Remark 8.

Another way how to prove the claim above is to directly verify that both λ1 and λ2 are the maximal CI-norms that are bounded from above on the set A by the prescribed values there. Since such a maximal CI-norm, if it exists, is unique, we get the equality between λ1 and λ2.

Operation 1.

Suppose that there are i<i′<n and some k≥1 such that

1. wi⁢…⁢wi′∈𝒥ρ,

2. wi′+1⁢…⁢wi′+k is a trivial subword,

3. i′+k+1∈Iρ, or i′+k+1∈Jρ in which case wi′+k+1 is the left-most letter of some Jm∈𝒥ρ.

Then we can “move” the trivial subword wi′+1⁢…⁢wi′+k in front of the subword wi⁢…⁢wi′ and change the match accordingly. That is, we get a new word

such that vj=wj for j<i and j>i′+k, vi-1+l=wi′+l, for l≤k, and vj+k=wj, for i≤j≤i′. The match ρ on w is changed accordingly in an obvious way to a match ν on v. That is, ν⁢(s)=t, for letters s,t in v, if and only if ρ⁢(s)=t, if we view the word v as a permutation of letters in w. Also 𝒥ρ is changed to 𝒥ν accordingly. Clearly, v′=w′=a and λ𝒥νν⁢(v)=λ𝒥ρρ⁢(w).

Operation 2.

Suppose that there are i<i′<n such that

1. wi⁢…⁢wi′∈𝒥ρ,

2. wi′+1=wi′-1 and i′+1∈Iρ.

Denote the subword wi⁢…⁢wi′-1 by V. Set j=ρ⁢(i′+1), therefore wj=wi′. We have either j>i′+1, or j<i′+1 (in which case necessarily j<i). We suppose the former, i.e. j>i′+1, the other case is analogous. We modify the word w to a word v by deleting the letters wi′ and wi′+1 from w and replacing the letter wj (=wρ⁢(i′+1)) by the subword V-1⁢V⁢wj (note that V⁢wj is equal to the word wi⁢…⁢wi′). That is, the original word w was

while the new word v is

Clearly, v′=w′. We must also change the match ρ into a match ν on v. We put {i,…,i′-1} into Iν as well as the indices corresponding to the subword V-1 of the subword V-1⁢V⁢wj, and we ‘connect them’ by the match ν; that is, for i≤l≤i′-1, ν⁢(wl)=wl′, where wl′ is the corresponding inverse letter to wl in the subword V-1. On the other hand, the subword V⁢wj of V-1⁢V⁢wj now belongs to 𝒥ν (previously, the equivalent subword wi⁢…⁢wi′ belonged to 𝒥ρ). On the rest, ν is defined the same way as ρ; that is, all other letters s,t from the word v can be identified with some letters in w, and we have ν⁢(s)=t if and only if ρ⁢(s)=t. It is clear that λ𝒥νν⁢(v)=λ𝒥ρ⁢(w).

Like Operation 1, Operation 2 also has its symmetric copy which we do not describe here.

Proof of Proposition 1.

Using Observation 5, we suppose that p1 and p2 are CI-norms. That is, we suppose p1 to be a finitely generated CI-norm on the free group on n generators {f1′,…,fn′} denoted by Fn′, generated by the values on some finite A′⊆Fn′, and p2 to be a finitely generated CI-norm on the free group on m generators {f1,…,fm} denoted by Fm, m≥n, generated by the values on some finite A⊆Fn. Recall from the statement of Proposition 1 that we also have that p1≥p2↾Fn and that p1 is δ-close to p2↾Fn, for some δ>0. We define a CI-norm p on Fn′∗Fm. Set D={(fi′)ε⋅fi-ε:i≤n,ε∈{-1,1}}. We let p to be generated by the values of p1 on Fn′, by the values of p2 on Fm and by the value δ on D. That is, for a∈Fn′∪Fm∪D we set

The norm p then extends to Fn′∗Fm using formula (2). It is clear that p is in fact finitely generated as it is generated by the values on A′∪A∪D.

For any i≤n, we have by definition p⁢(fi-1⋅fi′)≤δ, so we must check that p agrees with p1 on Fn′ and with p2 on Fm. The latter is easy and we show it first.

Take some a∈Fm and suppose that p⁢(a)<p2⁢(a). Then there are elements a1,…,ak∈A′∪A∪D such that a=a1⁢⋯⁢ak and ∑i=1kp⁢(ai)<p2⁢(a). Now for each i≤k, let ai′∈Fm be an element obtained from ai by replacing each generator fj′, resp. its inverse, j≤n, occurring in the element ai as a reduced word over the alphabet {f1′,…,fn′}±∪{f1,…,fm}±∪{1}, by the generator fj, resp. its inverse. Notice in particular that if ai∈D, then ai′=1. Since a∈Fm, we have that a=a1′⁢⋯⁢ak′. Note that we have p⁢(aj′)≤p⁢(aj), for all j≤k. Indeed, if aj∈A, then aj′=aj; if aj∈D, then aj′=1, and finally if aj∈A′, then since p1≥p2↾Fn we get that p⁢(aj′)≤p⁢(aj). It follows that

which is a contradiction.

So we focus on the former, i.e. that p agrees with p1 on Fn′.

Take some a∈Fn′ and suppose that p⁢(a)<p1⁢(a). Then there is some word w=w1⁢…⁢wk in the alphabet {f1′,…,fn′}±∪{f1,…,fm}±∪{1} such that w′=a and a match ρ on w, and a subdivision 𝒥 of Jρ such that p⁢(a)=p𝒥ρ. Set K={i≤k:wi∈Fm}. Let us define a graph structure on K. We declare i,j∈K to be connected by an edge

1. of type (1) if j is the least element such that i<j, wj=wi-1, and wi+1⁢…⁢wj-1 is a trivial word; or symmetrically, i.e. with the roles of i and j reversed,

2. of type (2) if i,j∈Iρ and ρ⁢(i)=j, ρ⁢(j)=i.

We remark that this graph is analogous to that one defined by Gao in [2, Theorem 2.6.5] when computing the Graev metric.

It follows, since the degree of each vertex is either 1 or 2 (note that no vertex i∈K can be isolated since the letter wi must get cancelled), that the connected components of K are either simple paths or cycles.

Next notice that if i∈K∩Jρ, then wi is a letter of a subword, denoted by V⁢(i), from 𝒥ρ. Either V⁢(i)∈Fm or V⁢(i)∈D. The paths in K that have exactly one end i with V⁢(i)∈D are called good paths, the others are called bad paths.

This assumption is what was taken for granted in the original proof of Proposition 1.15 in [1]. Let us show how to conclude the proof provided that Claim 12 is proved. Let us replace the word w=w1⁢…⁢wk by a word v=v1⁢…⁢vk, where, for i≤k,

1. vi=wi if wi∈Fn′,

2. vi=fj′⁣ε if wi=fjε∈Fm, for j≤n, ε∈{-1,1}.

We note that indeed necessarily j≤n, since i lies in a good path. The match on v is the same as on w, however we shall denote it by ν for notational reasons.

Now we still have v′=a, and by definition, p1⁢(a)≤p𝒥νν⁢(v). Thus we must show that p𝒥νν⁢(v)≤p𝒥ρρ⁢(w), i.e. ∑J∈𝒥νp⁢(J)≤∑J∈𝒥ρp⁢(J). Let ϕ be the isomorphism from Fn≤Fm to Fn′ which sends fi to fi′, for i≤j. Each J∈𝒥ρ from D was replaced by a trivial word in 𝒥ν. Let k be the number of such J’s. On the other hand, each J∈𝒥ρ that lies in Fn≤Fm was replaced by ϕ⁢(J) in 𝒥ν that lies in Fn′. Let k′ be the total length of such words. By our assumption that there are only good paths in K, we have k≥k′. Indeed, for each J∈𝒥ρ that lies in Fn and each subletter wi of J there is one good path with one end i and the other end being j with wj=wi-1, and V⁢(j)∈D.

Since p1 is δ-close to p2, we thus get that

and we are done. It remains to prove the claim.

This finishes the proof of Proposition 1. ∎