Exercise 4, § 4

Let G be a group and let A and B be subgroups of G. Let $\varphi : A\rightarrow B$ be an isomorphism.

Let $\boldsymbol{r}'$ be a family in the free group F(G) such that G is isomorphic to the group $F(G,\boldsymbol{r}')$ (A, I, p. 87, No. 6). The HNN extension of G according to $\varphi$ is by definition the group $G^{\varphi}= F(X,\boldsymbol{r})$ where X is the sum of the sets G, A, B and of an element $t$, and where $\boldsymbol{r}$ is the sum family of $\boldsymbol{r}'$ and of the relators $t^{-1}at=\varphi (a)$ for all $a\in A$.

We denote by $j: G\rightarrow G^{\varphi}$ the unique homomorphism such that $j(g)$ is the element represented by $g$ in $G^{\varphi}$.

a) Let $k\geqslant 1$ be an integer and let $(g_0, \varepsilon_1, g_1, . . . , \varepsilon_k, g_k)$ be a finite family such that $g_i\in G$ and $\varepsilon_i\in \{-1,1\}$ for all $i$. Suppose that there exists no integer $j$ such that $1\leqslant j\leqslant k-1$ and

or else

Then the element

of $G^{\varphi}$ is not the identity element ("Britton's Lemma".)

b) The homomorphism $j$ is injective.

c) Let $a$ and $b$ be distinct sets. Let G be the group $F(\{a, b\}, a^{-1}b^2ab^{-3})$. Put $b_1=a^{-1}ba$. Let H be a normal subgroup of finite index of G and $\pi : G\rightarrow G/H$ the canonical projection. One has $[b_1, b]\in$ Ker($\pi$ ).

d) The group G admits no faithful finite-dimensional representation. (Prove that G is not residually finite.)