Lecture 2

Stronger NFT. Choose right coset representatives for C\A, Ca_{1}, Ca_{2}, … and C\B, Cb_{1}, Cb_{2}, … This gives a unique normal form a_{0}b_{1}a_{2}b_{3} ⋯ or b_{0}a_{1}b_{2}a_{3} ⋯ with the property that each a_{i}, b_{i} is one of the coset representatives.

Remark. This is not good either. It rather should be ca_{0}b_{1}a_{2}b_{3} ⋯

Existence of “ordinary” normal forms: each g ∈ A \begin{matrix}∗ \\ C \end{matrix} B equals g_{1}g_{2} ⋯ g_{k}, g_{i} ∈ A ⊔ B. Continually “combine like terms” (b_{1}a_{2}a_{3}b_{4} ⟶ b_{1}(a_{2}a_{3})b_{4}) and combine elements of C with neighboring elements (a_{1}Φ_{b}(c)a_{2} → a_{1}Φ_{a}(c)a_{2} → (a_{1}Φ_{a}(c))a_{2}).

Existence of “stronger” normal forms: given an “ordinary” normal form g_{0}g_{1} ⋯ g_{3}g_{4} we continually replace g’s with suitable coset representatives, starting with g_{4}:

g_{0}g_{1} ⋯ g_{3}g_{4} ⟶^{g_{4}∈Ca_{4}} g_{0}g_{1} ⋯ (g_{3}c_{4})a_{4} ⟶^{g_{3}c_{4}∈Cb_{3}} g_{0}g_{1} ⋯ c_{3}b_{3}a_{4} → … → ca_{0}b_{1} ⋯ b_{3}a_{4}

HNN Extensions. The idea is to construct something given

\xymatrix{A & C \ar[l]_{Φ_{1}} \ar[l]^{Φ_{2}} & or & C_{1} \ar@{^{(}->}[dr] \ar[rr]^{Φ} &  & C_{2} \ar@{^{(}->}[dl] \\  &  &  &  & A}

Definition 1. The HNN extension (HNN stands for Higman–Neumann-Neumann) associated to this data is the group A∗_{C} or A∗_{Φ}

⟨A, t|c^{t} = Φ(c) ∀c ∈ C_{1}⟩ ≅ ^{A ∗ ⟨t⟩/} ⟨⟨c^{t}Φ(c^{−1}), c ∈ C_{1}⟩⟩.

t is called stable letter.

NFT for HNN Extensions.

Theorem 1. Every element in A∗_{C} can be expressed as a_{0}t^{ε_{1}}a_{1}t^{ε_{2}}a_{2}t^{ε_{3}} ⋯, each ε_{i} = ±1, each a_{i} ∈ A, and there is no appearance of t^{−1}c_{1}t, c_{1} ∈ C_{1} and tc_{2}t^{−1}, c_{2} ∈ C_{2}.

Length of g is number of t’s in its normal form. Length is invariant.

A imbeds in A∗_{C}.

Example. Let A = A_{5}, β = (12)(34), γ = (123). Then ⟨A, t|β^{t} = γ⟩ ≅ ℤ.

Hopfian groups.

Definition 2. A group G is hopfian if every epimorphism G → G is a monomorphism.

Exercise. Finitely generated abelian group is hopfian.

Exercise. ℤ^{∞}, F_{∞} are not hopfian. (Hint: kill one generator.)

Definition 3. BS (m, n) = ⟨a, t|(a^{m})^{t} = a^{n}⟩ (m, n ≠ 0). (BS stands for Baumslag– Solitar.)

\xymatrix{mℤ \ar@{^{(}->}[dr] \ar[rr]^{Φ=· \frac{n}{m}} &  & nℤ \ar@{^{(}->}[dl] \\  & ℤ}

G = BS (2, 3) is not hopfian. Consider homomorphism ψ : G → G induced by t ↦ t, a ↦ a^{2}. Then the relation [(a^{2})^{t} = a^{3}] ↦ [(a^{4})^{t} = a^{3}], so ψ is really a homomorphism. Further, ψ is surjective since t = ψ(t), a = ψ(a^{−1}(a)^{t}), but not injective, since 1 ≠ [a^{t}, a] maps to 1. Show that. [a^{t}, a] = t^{−1}a^{−1}ta^{−1}t^{−1}ata — is a normal form of length > 0. [a^{t}, a] ↦ [(a^{2})^{t}, a^{2}] = [a^{3}, a^{2}] = 1.

Reminder. x^{y} = y^{−1}xy, [x, y] = x^{−1}y^{−1}xy. Exercise. BS (1, n) is hopfian, BS (n, n) is hopfian. Constructions of amalgamated product and HNN extension are believed to be

“equally strong”. In the following example we present a group both as an amalgamated product and as a HNN extension.

Example. G = ⟨a, b, c, d|aba^{−1}b^{−1}cdc^{−1}d^{−1}⟩

• G ≅ F_{2} \begin{matrix}∗ \\ ℤ \end{matrix} F_{2} ≅ ⟨a, b⟩ \begin{matrix}∗ \\ ℤ \end{matrix} ⟨c, d⟩, (ℤ = ⟨[a, b]⟩ = ⟨[c, d]⟩).

• G ≅ F_{3}∗_{ℤ}, given by

\xymatrix{⟨b^{−1}cdc^{−1}d^{−1}⟩ \ar@{^{(}->}[dr] \ar[rr]^{φ:b^{−1}cdc^{−1}d^{−1}↦b^{−1}} &  & ⟨b⟩ \ar@{^{(}->}[dl] \\  & ⟨b, c, d⟩}

and stable letter a, that is (b^{−1}cdc^{−1}d^{−1})^{a} = b^{−1}.

A Graph of Groups. Let Γ be a directed graph. For each vertex v ∈ Γ, we have a group G_{v}. For each edge e ∈ Γ, we have group G_{e}. An edge is attached at its initial vertex i(e) and terminal vertex t(e).

G_{e} has monomorphisms to G_{i(e)}, G_{t(e)}.

φ_{e+} : G_{e} → G_{t(e)}, φ_{e−} : G_{e} → G_{i(e)}

From this data you can get a group π_{1}Γ, π_{1}(graph of groups). Consider all combinatorial paths starting at some fixed vertex of the graph (see

figure): abc, abe^{−1}d^{−1}, …

Equivalence relation is generated by removal of backtracks: abcc^{−1}e^{−1} ∼ abe^{−1}. So in our example, we get a subgroup ⟨a, b, c, d, e|−⟩ generated by ⟨abc, dec^{−1}⟩.

Label each edge a “stable” letter t_{i}, pick a base point. Our group is the set of equivalence classes of “words” of the form h_{0}s_{1}h_{1}s_{2}h_{2} ⋯ h_{k}s_{k}h_{k+1},

where each s_{i} = some t_{j}^{±1}.

If ht_{i}h′ is in the word, then h ∈ G_{i(e_{i})}, h′ ∈ G_{t(e_{i})}; if h′t_{i}^{−1}h is in the word, then h ∈ G_{i(e_{i})}, h′ ∈ G_{t(e_{i})}. Equivalence relation is generated by “backtracks”:

φ_{i−}(g_{i})t_{i} ∼ t_{i}φ_{i+}(g_{i}) ∀g_{i} ∈ G_{i} = G_{e_{i}}.