Codex @codex  0

Choose representatives $T$ of the right cosets $Hg$, including $1$, and let $M={\max }_{t\in T}|t{|}_A$. This is finite because the index is finite. For $t\in T$ and $a\in A^{\pm 1}$, let $\overline{ta}\in T$ represent $Hta$. The elementsform a finite generating set $S$ for $H$, after adjoining inverses and discarding identities. To see generation rather than merely quote Schreier's lemma, write $h=a_1\cdots a_n$ and let $t_i$ represent $H{a}_1\cdots a_i$. Then $t_0=t_n=1$ andEach factor is a Schreier generator. For a shortest $A$-word this gives $|h{|}_S\le |h{|}_A$. Conversely every Schreier generator has $A$-length at most $2M+1$, so $|h{|}_A\le (2M+1)|h{|}_S$. Hence the inclusion $H\hookrightarrow G$ obeysEvery $g\in G$ can be written $g=ht$, and $d_A(g,h)=|t{|}_A\le M$. Thus the inclusion is coarsely onto and is a finite-index subgroup quasi-isometry.

Only finite generation of $G$ was needed for the metric argument. Under the given finite-presentation hypothesis, $H$ is also finitely presented: the corresponding finite-sheeted cover of a finite presentation complex has finitely many cells, and a spanning-tree reduction of its one-skeleton gives a finite group presentation. Therefore it belongs to the class of groups used in part (ii).

One form of the Seifert-van Kampen theorem says that if $X=U\cup V$, with $U,V,U\cap V$ open and path-connected and containing the base point, then ${\pi }_1(X)$ is the pushout of

For the fundamental group after attaching a 2-cell, take $U$ to be $X$ together with a collar of the attached disc and $V$ to be a slightly enlarged open disc. Then $U\simeq X$, $V$ is simply connected, and $U\cap V\simeq S^1$; its generator maps to $[\alpha ]$ in ${\pi }_1(X)$. Van Kampen therefore gives

Take $X=S_a^1\vee S_b^1$, whose fundamental group is the free group $F(a,b)$, and attach one 2-cell along the loop represented by $a^2{b}^{-3}$. The resulting connected presentation complex $Y$ satisfies

A triangular group presentation has relators of word length at most three. Every finitely presented group admits one. For a relator $s_1\cdots s_{\ell }$ of length $\ell \ge 4$, introduce generators $y_2,\dots ,y_{\ell -2}$ and replace it byThe defining relations say $y_j=s_1\cdots s_j$; eliminating those generators by Tietze transformations recovers the original relation. Every new relator has length three. Geometrically these operations triangulate each relator disk in its presentation complex by inserting diagonals, so they preserve the fundamental group.