# Presentations of Amalgamated Free Products of Two Groups.

Suppose we have an amalgamated free product $H\ast_LK$ of groups $H$ and $K$ with respect to a (normal) subgroup $L$ (of both $H$ and $K$), where $H\equiv\langle X\mid R\rangle$ and $K\equiv\langle Y\mid S\rangle$.

What is a presentation for $H\ast_LK$ in terms of $\langle X\mid R\rangle$ and $\langle Y\mid S\rangle$?

I've looked in (the old version of) "Presentation (sic) of Groups," by D. L. Johnson; "Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations," by Magnus et al.; and "Combinatorial Group Theory," by Lyndon et al. It doesn't seem to be anywhere obvious online.

My main issue is that I'm not working with a definition of amalgamated free products of groups, only a vague understanding; the definition on Wikipedia (in the generalisation section of the article on free products) is satisfactory, I guess, but I could do with an equivalent definition (if there be such) from a combinatorial group theoretic perspective rather than a categorical one - and I think answering the question here will provide one. (Thus I've added the definition tag.)

I don't have a copy of the latest "Presentation$\color{red}{s}$ of Groups," by D. L. Johnson but, apparently, it's in there somewhere, so please don't just cite the book unless there's a Google books page of the presentation or something like that.

• This is definitely in lyndon and schupp! So then I don't understand your question. Did you not understand their definition? Or could you not find it in there? (Their definition is the "standard" combinatorial one.) – user1729 Apr 22 '18 at 20:35
• @user1729: No, I couldn't find it! Let me check again. Give me 20 minutes, please. – Shaun Apr 22 '18 at 20:37
• (If $H$ and $K$ have isomorphic subgroups $L_1\leq H$, $L_2\leq K$ with isomorphism $\varphi:L_1\rightarrow L_2$ then the free product of $H$ and $K$ amalgamating $L$ has relative presentation $\langle H, K:\varphi(h)=h\rangle$. The sub group $L$ is the image of $L_1$ and $L_2$ under this "pinning together". – user1729 Apr 22 '18 at 20:41
• (I don't have my copy of lyndon and schupp with me, but it is in a section on "embedding theorems", if I recall correctly.) – user1729 Apr 22 '18 at 20:44
• (Also, note the word "relative" in my above comment. For example, free products with amalgamation do not actually have a normal form. Rather, they have a "relative" normal form.) – user1729 Apr 22 '18 at 20:46

A presentation for $H*_LK$ is $\langle X,Y\mid R,S,T\rangle$ where $T$ is as follows. For each $\ell\in L$, choose words $x_\ell$ and $y_\ell$ representing $\ell$ using generators from $X$ and $Y$, respectively. Then $T$ is the set of the words $x_\ell y_\ell^{-1}$ for all $\ell\in L$ (or, it suffices to just take a collection of $\ell$s that generate $L$). In other words, for each element (or generator) of $L$, we add a relation saying that its representation in $H$ is the same as its representation in $K$.
(More precisely, it is rather rare that $L$ is actually literally a subgroup of both $H$ and $K$. Rather, we have a subgroup $L$ of $H$, a subgroup $L'$ of $K$, and an isomorphism $f:L\to L'$. Then we would take $T$ to consist of words $x_\ell y^{-1}_\ell$ where $x_\ell$ is a word representing $\ell\in L$ and $y_\ell$ is a word representing $f(\ell)\in L'$.)
For a simple example, suppose $H$ is cyclic of order $4$, $K$ is cyclic of order $6$, and we are amalgamating them over their subgroup of order $2$. Then $\langle x\mid x^4\rangle$ is a presentation of $H$ and $\langle y\mid y^6\rangle$ is a presentation of $K$. A generator of $L$ is $x^2$ in $H$ and $y^3$ in $K$. So, we obtain the presentation $\langle x,y\mid x^4,y^6,x^2y^{-3}\rangle$ for $H*_L K$.