Let $F_n$ be the free group on $n$ generators. Of course, every finitely-presentable group $G$ is a finite colimit of copies of $F_n$, where $n$ is allowed to vary. But is $G$ a finite colimit of copies of $F_2$?

Of course, because $F_{2n}$ is a finite coproduct of copies of $F_2$, we have that any finitely-presentable group $G$ is a finite colimit of finite colimits of copies of $F_2$ -- a "2-fold" finite colimit of copies of $F_2$. But I'm curious about the 1-fold case.

To make the question a bit more concrete, let's unwind what it means to be a finite colimit of copies of $F_2$:

Let $G$ be a group. Then $G$ is a finite colimit of copies of copies of $F_2$ if and only if $G$ admits a presentation of the following description:

  • There are $2n$ generators coming in pairs $x_1,y_1, \dots x_n, y_n$;

  • There is a finite set of generating relations, each of the form $w(x_i,y_i)=v(x_j,y_j)$, where $w,v$ are group words and $1 \leq i \leq j \leq n$.

So for example, $x_1y_1^2x_1^{-1} = y_2^{-1}x_2$ is a permissible generating relation (with $i=1,j=2$) but $x_1 x_2 = x_3$ is not a permissible generating relation because only 2 different subscripts are allowed to appear in a permissible generating relation. So my question is:

Question: Let $G$ be a finitely-presented group.

  • Is $G$ a finite colimit of copies of $F_2$?

  • Equivalently, does $G$ admit a presentation of the above form?


The form of the presentation can be constrained even further, to look like this:

  • There is a finite set of generating relations, coming in pairs each of the form $x_i = w(x_j,y_j)$, $y_i=v(x_j,y_j)$, where $w,v$ are group words and $1 \leq i, j \leq n$.

Other variations are possible too; I'm not sure what the most convenient description to work with might be.

  • $\begingroup$ Here is how to present your illegal relation $x_1 x_2 x_3 = 1$ in a legal way. That group has $2n$-presentation $\langle x_1, y_1, x_2, y_2, x_3, y_3 \mid x_1 x_2 x_3, y_1, y_2, y_3 \rangle$. To make $x_1 x_2 x_3$ into a legal relation, introduce $x_4,y_4, x_5, y_5$, and relations $x_1 = x_4, x_2 = y_4$, $x_5 = x_4 y_4$, $x_3 = y_3$; then the legal presentation is $$\langle x_1, \cdots, y_5 \mid x_1 = x_4, x_2 = y_4, x_3 = y_3, y_1 = y_2 = y_3 = 1, x_4 y_4 = x_5\rangle.$$ This seems to lead to the algorithm in Pace Nielsen's answer. $\endgroup$ – Mike Miller May 27 at 20:56
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    $\begingroup$ @MikeMiller Thanks for pointing that out! (And yes, that was exactly what motivated the solution I produced.) $\endgroup$ – Pace Nielsen May 27 at 21:01

I believe the answer is yes. Assume, by way of contradiction, that some finitely presented group cannot be so expressed. Then we can choose such a group $G$ where for any generating set of the form $x_1,y_1, x_2,y_2,\ldots, x_n,y_n$ the number of non-permissible relations needed to define $G$ (together with some finite number of permissible relations) is minimized; say those non-permissible relations are $w_1=1, w_2=1,\ldots, w_m=1$. Write $w_1=z_1z_2\cdots z_p$ where each $z_j\in \{x_1^{\pm 1},y_1^{\pm 1},\ldots, x_n^{\pm 1},y_n^{\pm 1}\}$, where we may also assume that $p$ has been minimized. Note that since $w_1=1$ is not permissible, we must have $p\geq 3$.

Add new generators $x_{n+1},y_{n+1},x_{n+2},y_{n+2}$. The relations $x_{n+1}=z_{p}$, $y_{n+1}=z_{p-1}$, $x_{n+2}y_{n+2}=1$ are permissible. The relation $x_{n+2}y_{n+1}x_{n+1}=1$ is also permissible (since it is equivalent to $x_{n+2}=x_{n+1}^{-1}y_{n+1}^{-1}$). Asserting these relations still gives us the same group (since our new relations merely tell us how to write the new generators in terms of the old ones). The relation $w_1=1$ is equivalent to $z_1z_2\cdots z_{p-2}y_{n+2}=1$, but it is now shorter, a contradiction.

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    $\begingroup$ Ah, you solved my notational problem (this may be standard). I incorporated it as well to make my answer more readable. $\endgroup$ – R. van Dobben de Bruyn May 27 at 21:19
  • $\begingroup$ Thanks! Incidentally, I think this algorithm generalizes to show that for any variety (in the sense of universal algebra) with all operations of arity $\leq n$, any finitely-presented algebra is a finite colimit of copies of the free algebra on $n$ generators. I wonder if this is known and written somewhere... $\endgroup$ – Tim Campion May 27 at 21:20
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    $\begingroup$ If only you did not unecessarily say "proof by contradiction", you would have an algorithm in your hands for converting any presentation to a permissible one. $\endgroup$ – Andrej Bauer May 28 at 6:12
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    $\begingroup$ @TimCampion Semigroups are even easier. Suppose you have a relation $x_1 x_2 \cdots x_m = y_1 y_2 \cdots y_n$. Create new elements $z_1,z_2,\ldots, z_{m-1}$ and $w_1,w_2,\ldots, w_{n-1}$ and add the relations $z_1=x_1x_2$, $z_2=z_1x_3$, $z_3=z_2x_4$, and so forth up to $z_{m-1}=z_{m-2}x_m$. Do the something similar for the $w$'s. This defines the $z$ and $w$ variables in terms of previous generators. Finally, asserting $z_{m-1}=w_{n-1}$ gives the original relation. $\endgroup$ – Pace Nielsen Sep 3 at 18:51
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    $\begingroup$ @TimCampion The word problem for groups is not decidable. In particular, one cannot tell whether a group word is trivial (i.e., equal to $1$). But, given any finite presentation, the algorithm can still easily be made effective. Rather than working by contradiction, one instead just reduces word lengths (ala what Andrej Bauer said earlier). $\endgroup$ – Pace Nielsen Sep 3 at 18:59

Here's a pretty direct way to do this. Choose any presentation by generators $x_1,\ldots,x_n$ and relations $r_1,\ldots,r_m$; say $r_i = z_{i,1} \cdots z_{i,k}$ for $z_{i,1},\ldots,z_{i,k} \in \{x_1^{\pm 1}, \ldots, x_n^{\pm 1}\}$. Firstly, we may assume all $r_i$ have length $k = 3$: the new variables $$x_{i,j} = z_{i,1} \cdots z_{i,j}$$ for $0 \leq j \leq k$ are subject only to the relations \begin{align*} x_{i,0} = e = x_{i,k}, & & & & & & x_{i,j} = x_{i,j-1}z_{i,j} & & (1 \leq j \leq k). \end{align*} If $k < 3$, we can eliminate the variable $z_{i,1}$ at the expense of replacing all $z_{i,1}^{\pm 1}$ by $z_{i,2}^{\mp 1}$ (if $k = 2$) or $e$ (if $k = 1$) in the other relations, so we may assume all $r_i$ have length exactly $3$. Then introduce new variables $x_{n+1},\ldots,x_{n+m}$ as well as variables $y_1,\ldots,y_{n+m}$, subject to the relations \begin{align*} x_{n+i} &= z_{i,1},& & 1 \leq i \leq m,\\ y_i &= e & & 1 \leq i \leq n,\\ y_{n+i} &= z_{i,2}, & & 1 \leq i \leq m,\\ x_{n+i}y_{n+i} &= z_{i,3}^{-1}, & & 1 \leq i \leq m,\\ \end{align*} This gives a presentation of the desired form. $\square$

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  • $\begingroup$ Thanks! It's interesting to see this from a more top-down perspective. $\endgroup$ – Tim Campion May 27 at 21:21

Ok, I think the above answers have pointed the way to a proof of the obvious generalization:

Theorem: Consider a variety $V$ (in the sense of universal algebra) generated by operations of arity bounded by some $N \in \mathbb N$, and let $\mathcal C$ be the category of $V$-algebras and homomorphisms. Let $F$ be the free algebra on $N$ generators. Then every finitely-presented $A \in \mathcal C$ is a finite colimit of copies of $F$.

Proof: By using dummy variables, we may assume that every basic operation in $V$ is of arity exactly $N$. As described in the case of groups in the Question, we are looking for a presentation of $A$ by generators $(x_{11}, \dots, x_{1N},\dots, x_{n1}, \dots, x_{nN})$ modulo "permissible" relations $w(x_{i 1}, \dots x_{i N}) = v(x_{j 1},\dots, x_{j N})$, were $w, v$ are (possibly composite) operations in the variety $V$.

As in Pace Nielsen's answer, it will suffice to show that if $w(x_{11},\dots, x_{nN}) = v(x_{11},\dots,x_{nN})$ is a non-permissible relation, we can, after adding more variables $(x_{n+1,1},\dots, x_{n+1,N})$, replace it with relations using only shorter words (in the sense that the total number of basic operations of which each word is composed is smaller -- the base case is a word $x_{ij}$ composed of no operations; note that a relation $x_{ij} = x_{kl}$ is permissible) and a relation of the form $f(x_{n+1,1},\dots, x_{n+1,N}) = v(x_{11},\dots, x_{nN})$.

To this end, we may write $w(x_{11},\dots, x_{nN}) = f(w_1(x_{11},\dots,x_{nN}), \dots, w_N(x_{11},\dots,x_{nN}))$ where $f$ is a basic operation and the $w_i$'s are shorter words. We impose the relations $x_{n+1,i} = w_i(x_{11},\dots,x_{nN})$ (which use only shorter words) along with the relation $f(x_{n+1,1},\dots, x_{n+1,N}) = v(x_{11},\dots, x_{nN})$, as desired.

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  • $\begingroup$ I think the argument generalizes, with basically just notational changes, to the case of a many-sorted variety with finitely many sorts (such as the 2-sorted variety of (ring, module) pairs -- showing that any such pair which is finitely presentable is a finite colimit of copies of $\mathbb Z[x,y]$ acting on itself). $\endgroup$ – Tim Campion Sep 4 at 15:19

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