3.1 A geometric version of Theorem 1.1

We remark that Theorem 1.1 can be slightly modified as to avoid a representation $\rho ~:~G \twoheadrightarrow \mathfrak {X}(R) \rtimes \mathcal {H}(R)$ with sequence $\ker (\rho ) \hookrightarrow G \twoheadrightarrow \mathfrak {X}(R) \rtimes \mathcal {H}(R)$ split. We shall relax the hypothesis on $\rho $ at the cost of an assumption on the base ring R.

Recall that a metric space Y is a quasi-retract of a metric space X if there exists a pair $r : X \to Y$ and $\iota : Y \to X$ of $(C,D)$ -Lipschitz functions such that $d_Y(r \circ \iota (y), y) \leq D$ for all $y \in Y$ ; see [Reference Alonso6]. The point now is that quasi-retracts also inherit homotopical finiteness properties. In particular, if G and H are finitely generated groups such that $r : G \to H$ is a quasi-retract, Alonso proved that $\phi (G) \leq \phi (H)$ ; see [Reference Alonso6, Theorem 8]. Of course, group retracts are particular examples of quasi-retracts.

The advantage here is that a quasi-retract $r : G \to H$ needs not be a group homomorphism. In a recent remarkable paper, Skipper et al. [Reference Skipper, Witzel and Zaremsky52] used the finiteness length and quasi-retracts to construct infinitely many quasi-isometry classes of finitely presented simple groups.

Using the geometric language above, we have the following.

4.1 A space for $\textbf {A}_n(R)$

Recall that a covering of a set X is a collection of subsets $\left \{ X_{\lambda } \right \}_{\lambda \in \Lambda }$ of X whose union is the whole of X, i.e., $X = \cup _{\lambda \in \Lambda } X_{\lambda }$ . The nerve of the covering $\left \{ X_{\lambda } \right \}_{\lambda \in \Lambda }$ is the simplicial complex ${N(\left \{ X_{\lambda } \right \}_{\lambda \in \Lambda })}$ defined as follows. Its vertices are the sets $X_{\lambda }$ for $\lambda \in \Lambda $ , and $k+1$ vertices $X_{\lambda _0}, X_{\lambda _1}, \ldots , X_{\lambda _k}$ span a k-simplex whenever the intersection of all such $X_{\lambda _i}$ is nonempty, i.e., $\cap _{i=0}^k X_{\lambda _i} \neq \varnothing $ .

In [Reference Abels and Holz4, Reference Holz39], Holz and Abels investigate nerve complexes attached to groups as follows. Fixing a family of subgroups, they take the nerve of the covering of the group by all cosets of subgroups of the given family. (Such spaces are also called coset posets or coset complexes in the literature.) More precisely, given a group G and a family $\mathscr{H} = \left \{ H_{\lambda } \right \}_{\lambda \in \Lambda }$ of subgroups of G, let $\mathfrak {H}$ denote the covering $\mathfrak {H} = \left \{ gH \mid g \in G, \, H \in \mathscr{H}\;\right \}$ of G by all left cosets of all members of $\mathscr{H}$ . The coset complex ${CC(\mathscr{H})}$ is defined as the nerve of the covering $N(\mathfrak {H}\;)$ . In particular, if the family $\mathscr{H}$ is finite, then $CC(\mathscr{H})$ is $(\vert \mathscr{H}\;\vert -1)$ -dimensional.

Inspiration for the above came primarily from the theory of buildings: if G is a group with a BN-pair $(G,B,N,S)$ , then the coset poset $CC(\mathscr{H}\;)$ associated to the family $\mathscr{H}$ of all maximal standard parabolic subgroups of G is by definition the building $\Delta (G,B)$ associated to the system $(G,B,N,S)$ ; see [Reference Abramenko and Brown5, Section 6.2] and [Reference Brück19]. As it turns out, coset complexes show up in many other contexts, such as Deligne complexes [Reference Charney and Davis30], Bass–Serre theory [Reference Serre51], $\Sigma $ -invariants of right-angled Artin groups [Reference Meier, Meinert and VanWyk44], and higher generating families of braid groups [Reference Bux, Baake, Götze and Hoffmann23, Reference Bux, Fluch, Marschler, Witzel and Zaremsky24] and automorphism groups of free groups [Reference Brück20, Reference Brück19].

Since vertices of $CC(\mathscr{H}\;)$ are cosets in the group G, it follows that G acts naturally on $CC(\mathscr{H}\;)$ by cell-permuting homeomorphisms, namely the action induced by left multiplication on the cosets $gH$ for $g \in G$ and $H \in \mathscr{H}$ .

Going back to Abels’ groups, consider the following $\mathbb {Z}$ -subschemes of $\textbf {A}_n$ .

$$ \begin{align*} H_1 = \left( \begin{smallmatrix} 1 & * & \cdots & * & 0 \\ 0 & * & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & * & \vdots \\ 0 & & 0 & * & 0 \\ 0 & 0 & \cdots & 0 & 1 \end{smallmatrix} \right), \, H_2 = \left( \begin{smallmatrix} 1 & 0 & \cdots & \cdots & 0 \\ 0 & * & * & \cdots & * \\ \vdots & \ddots & \ddots & * & \vdots \\ 0 & & 0 & * & * \\ 0 & 0 & \cdots & 0 & 1 \end{smallmatrix} \right), \, H_3 = \left( \begin{smallmatrix} 1 & * & 0 & \cdots & 0 \\ 0 & * & 0 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 & 0 \\ 0 & & 0 & * & * \\ 0 & 0 & \cdots & 0 & 1 \end{smallmatrix} \right). \end{align*} $$

For $n = 4$ , we consider, in addition, the following subscheme.

$$ \begin{align*} H_4 = \begin{pmatrix} 1 & 0 & * & 0 \\ 0 & * & * & * \\ 0 & 0 & * & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \leq \textbf{A}_4. \end{align*} $$

The unipotent radicals of the matrix groups above—i.e., the intersections of each $H_i$ with the group $\textbf {U}_n$ of upper unitriangular matrices—are examples of group schemes arising from (maximal) contracting subgroups. Indeed, consider the locally compact group $\textbf {A}_n(\mathbb {K})$ for $\mathbb {K}$ a nonarchimedean local field. In this case, each unipotent radical $\mathcal {U}_i(\mathbb {K}) = H_i(\mathbb {K}) \cap \textbf {U}_n(\mathbb {K})$ is the contracting subgroup associated to the automorphism given by conjugation by some element t contained in the torus $\textbf {T}_n(\mathbb {K})$ ; see [Reference Abels2, Reference Baumgartner and Willis7, Reference Holz39]. Holz shows [Reference Holz39, Sections 2.7.3 and 2.7.4] that this defines a unipotent group scheme over $\mathbb {Z}$ depending on $t \in \textbf {T}_n(\mathbb {K})$ . Following Abels, we call the schemes $H_i$ above horospherical and their unipotent radicals $\mathcal {U}_i = H_i \cap \textbf {U}_n$ contracting subgroups of $\textbf {A}_n$ .

For R a commutative ring with unity and $n \geq 4$ , let $\mathscr{H}\;(n, R)$ denote the family of groups of R-points of horospherical subgroups of $\textbf {A}_n(R)$ , i.e.,

$$ \begin{align*} {\mathscr{H}\;(n,R)} = \begin{cases} \left\{ H_1(R), \, H_2(R), \, H_3(R), \, H_4(R) \right\}, & \text{if } n = 4;\\ \left\{ H_1(R), \, H_2(R), \, H_3(R) \right\} & \text{otherwise}. \end{cases} \end{align*} $$

We also let

$$ \begin{align*} \mathfrak{H}(n,R) = \left\{ gH \mid g \in \textbf{A}_n(R), H \in \mathscr{H}\;(n,R) \right\}. \end{align*} $$

In the notation above, the space we shall consider is the nerve complex

$$ \begin{align*} CC(\mathscr{H}\;(n,R)) = N(\mathfrak{H}(n,R)) \end{align*} $$

associated to the covering of $\textbf {A}_n(R)$ by the left cosets $\mathfrak {H}(n,R)$ of the horospherical subgroups listed above. As mentioned previously, the group $\textbf {A}_n(R)$ acts on the simplicial complex $CC(\mathscr{H}\;(n,R))$ by cell-permuting homeomorphisms via left multiplication.

The space $CC(\mathscr{H}\;(n,R))$ has many useful features. Some of the facts we are about to list here actually hold more generally for arbitrary coset complexes. In the case of the following lemma, the properties that are particular to our groups come from the facts that the chosen family $\mathscr{H}\;(n,R)$ is finite and that the group $\textbf {A}_n(R)$ is a split extension $\textbf {A}_n(R) = \textbf {U}_n(R) \rtimes \textbf {T}_n(R)$ such that the contracting subgroups $\mathcal {U}_i(R) = H_i(R) \cap \textbf {U}_n(R)$ (as well as intersections of contracting subgroups) are all $\textbf {T}_n(R)$ -invariant.

Proof Since the intersection of cosets in a group is a coset of the intersection of the underlying subgroups, it follows that $CC(\mathscr{H}\;(n,R))$ is homogeneous . That is to say, every simplex is contained in a simplex of dimension $k = \vert \mathscr{H}\;(n,R) \vert -1$ and every maximal simplex has dimension exactly k. (Note that $CC(\mathscr{H}\;(n,R))$ is even a chamber complex if $n \geq 5$ .) We observe that $CC(\mathscr{H}\;(n,R))$ is colored, with types (or colors) given precisely by the family of subgroups $\mathscr{H}\;(n,R)$ . Also, the given action of $\textbf {A}_n(R)$ on $CC(\mathscr{H}\;(n,R))$ is type-preserving and transitive on the set of maximal simplices. Thus, the maximal simplex given by the intersection

$$ \begin{align*}\bigcap\limits_{H \in \mathscr{H}\;(n,r)} H \end{align*} $$

is a fundamental domain for the $\textbf {A}_n(R)$ -action. In particular, since $\vert \mathscr{H}\;(n,R)\vert $ is finite, it follows that the action of $\textbf {A}_n(R)$ is cocompact.

The stabilizers of the $\textbf {A}_n(R)$ -action are also not difficult to determine. For instance, given a maximal simplex $\sigma = \left \{ g_1 H_1(R), \, g_2 H_2(R), \, g_3 H_3(R) \right \}$ in $CC(\mathscr{H}\;(n,R))$ with $n \geq 5$ , there exists $g \in \textbf {A}_n(R)$ such that $\sigma = g \cdot \left \{ H_1(R), \, H_2(R), \, H_3(R) \right \}$ . A group element $h \in \textbf {A}_n(R)$ fixes $\sigma $ if and only if $h \in g \cdot (H_1(R) \cap H_2(R) \cap H_3(R)) \cdot g^{-1}$ . A similar argument shows that a cell-stabilizer of $CC(\mathscr{H}\;(n,R))$ for any $n \geq 4$ is a conjugate of some intersection of subgroups that belong to the family $\mathscr{H}\;(n,R)$ .▪

The existence of a single simplex as fundamental domain, and the fact that cell-stabilizers are conjugates of finite intersections of members of a fixed family of subgroups, are properties that characterize coset complexes; see e.g., [Reference Bux, Fluch, Marschler, Witzel and Zaremsky24, Observation A.4 and Proposition A.5].

Having determined the cell-stabilizers, we now prove that they are finitely presented whenever we need them to be.

Proof We shall prove that, under the given assumption, the vertex-stabilizers are finitely presented. It will be clear from the arguments below that the same holds for stabilizers of higher dimensional cells. We remind the reader that the multiplicative group $\mathbb {G}_m(R)$ , which is a retract of $\textbf {B}_2^{\circ }(R)$ , is finitely presented because $\textbf {B}_2^{\circ }(R)$ itself is so; cf. Section 2. In particular, $\textbf {T}_n(R)$ is also finitely presented.

By Lemma 4.2, we need only show that the members of $\mathscr{H}\;(n,R)$ are finitely presented. By the commutator (2.1) and diagonal relations (2.2), we see that the “last-column subgroup” $\mathcal {C}_{n-1}$ of $H_1$ , given by

$$ \begin{align*} \mathcal{C}_{n-1} = \left( \begin{smallmatrix} 1 & 0 & \cdots & 0 & * & 0 \\ 0 & 1 & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots & \vdots \\ \vdots & & \ddots & 1 & * & \vdots \\ & & & 0 & * & 0 \\ 0 & \cdots & & \cdots & 0 & 1 \end{smallmatrix} \right), \end{align*} $$

is normal in $H_1$ . The quotient $H_1 / \mathcal {C}_{n-1}$ is isomorphic to the subgroup

$$ \begin{align*} Q_{n-1} = \left( \begin{smallmatrix} 1 & * & \cdots & * & 0 & 0 \\ 0 & * & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & * & \vdots & \vdots \\ \vdots & & \ddots & * & 0 & \vdots \\ & & & 0 & 1 & 0 \\ 0 & \cdots & & \cdots & 0 & 1 \end{smallmatrix} \right) \leq H_1. \end{align*} $$

We claim that the column subgroup $\mathcal {C}_{n-1}$ is itself finitely presented. To check this, let us briefly recall some concepts from the $\Sigma $ -theory of Bieri–Strebel [Reference Bieri, de Cornulier, Guyot and Strebel13]. Given an abelian group Q and a homomorphism $v : Q \to (\mathbb {R},+)$ , denote by $Q_v$ the monoid $Q_v = \left \{ q \in Q \mid v(q) \geq 0 \right \}$ . We say that a $\mathbb {Z}[Q]$ -module A is tame when A is finitely generated over $\mathbb {Z}[Q_v]$ or over $\mathbb {Z}[Q_{-v}]$ (possibly both) for every homomorphism $v : Q \to (\mathbb {R},+)$ . One of the main results of [Reference Bieri, de Cornulier, Guyot and Strebel13] states that a finitely generated group $A \rtimes Q$ (with A and Q abelian) is finitely presented if and only if A is a tame $\mathbb {Z}[Q]$ -module.

Going back to our proof, since $\textbf {B}_2^{\circ }(R)$ is assumed to be finitely presented, then so is the group

$$ \begin{align*}\mathbb{A}\text{ff}_-(R) \cong \mathbb{G}_a(R) \rtimes_{-1} \mathbb{G}_m(R) \cong \left( \begin{smallmatrix} 1 & * \\ 0 & * \end{smallmatrix} \right) \leq \operatorname{\textrm{GL}}_2(R)\end{align*} $$

by Lemma 3.2. In particular, the $\mathbb {Z}[\mathbb {G}_m(R)]$ -module $\mathbb {G}_a(R)$ with the given action

$$ \begin{align*} \mathbb{G}_m(R) \times \mathbb{G}_a(R) \ni (u,r) \mapsto u^{-1} r \end{align*} $$

is tame by [Reference Bieri, de Cornulier, Guyot and Strebel13, Theorem 5.1]. But $\mathcal {C}_{n-1}$ is seen to be isomorphic to $\mathbb {G}_a(R)^{n-2} \rtimes \mathbb {G}_m(R)$ , where the action of $\mathbb {G}_m(R)$ on each copy of $\mathbb {G}_a(R)$ is the reverse multiplication as above, and the action $\mathbb {G}_m(R) \curvearrowright \mathbb {G}_a(R)^{n-2}$ is just the diagonal action. Thus, by [Reference Bieri, de Cornulier, Guyot and Strebel13, Proposition 2.5(i)] it follows that $\mathbb {G}_a(R)^{n-2}$ is also a tame $\mathbb {Z}[\mathbb {G}_m(R)]$ -module, whence one has—again by [Reference Bieri, de Cornulier, Guyot and Strebel13, Theorem 5.1]—that $\mathcal {C}_{n-1} \cong \mathbb {G}_a(R)^{n-2} \rtimes \mathbb {G}_m(R)$ is finitely presented.

We have shown that $H_1$ fits into a (split) short exact sequence

$$ \begin{align*} \mathcal{C}_{n-1} \hookrightarrow H_1 \twoheadrightarrow Q_{n-1}, \end{align*} $$

where $\mathcal {C}_{n-1}$ is finitely presented. Decomposing $Q_{n-1}$ similarly via the last column, as we did with $H_1$ , a simple induction on n shows that $Q_{n-1}$ is also finitely presented. Lemma 2.1(1) thus shows that $H_1$ is finitely presented.

By considering the “first-row subgroup”

$$ \begin{align*} \mathcal{R}_{n-1} = \left( \begin{smallmatrix} 1 & 0 & & \cdots & & 0 \\ 0 & * & * & \cdots & * & * \\ \vdots & 0 & 1 & 0 & 0 & 0 \\ \vdots & & \ddots & \ddots & \ddots & \vdots \\ & & & 0 & 1 & 0 \\ 0 & \cdots & & \cdots & 0 & 1 \end{smallmatrix} \right) \end{align*} $$

of $H_2$ , which is also normal by (2.1) and (2.2), an argument analogous to the previous one, now using $\mathbb {A}\text {ff}(R) \cong \mathbb {G}_a(R) \rtimes \mathbb {G}_m(R) \cong \left ( \begin {smallmatrix} * & * \\ 0 & 1 \end {smallmatrix} \right ) \leq \operatorname {\textrm {GL}}_2(R)$ , shows that $H_2$ is also finitely presented.

The case of $H_3$ is more straightforward since it equals the direct product

$$ \begin{align*} \left( \begin{smallmatrix} 1 & * & 0 & \cdots & 0 & 0 \\ 0 & * & 0 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & \vdots & \vdots \\ \vdots & & \ddots & * & 0 & \vdots \\ & & & 0 & 1 & 0 \\ 0 & \cdots & & \cdots & 0 & 1 \end{smallmatrix} \right) \times \left( \begin{smallmatrix} 1 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 1 & \ddots & \vdots & \vdots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & \vdots \\ & & & 0 & * & * \\ 0 & \cdots & & \cdots & 0 & 1 \end{smallmatrix} \right) \cong \mathbb{A}\text{ff}_-(R) \times \mathbb{G}_m(R)^{n-4} \times \mathbb{A}\text{ff}(R) \end{align*} $$

and all factors on the right-hand side are finitely presented; see Lemma 3.2 and Section 2.

Establishing finite presentability of $H_4 \leq \textbf {A}_4(R)$ is slightly different. Consider the subgroups

$$ \begin{align*} \Gamma_1 = \left( \begin{smallmatrix} 1 & 0 & * & 0 \\ 0 & * & * & 0 \\ 0 & 0 & * & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix} \right) \text{ and } \Gamma_2 = \left( \begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & * & 0 & * \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix} \right) \end{align*} $$

and let $p_1 : \Gamma _1 \twoheadrightarrow Q$ and $p_2 : \Gamma _2 \twoheadrightarrow Q$ denote the natural projections onto the diagonal subgroup

$$ \begin{align*} Q = \left( \begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & * & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix} \right). \end{align*} $$

With this notation, we have that $H_4$ is isomorphic to the fiber product

$$ \begin{align*} P = \left\{ (g,h) \in \Gamma_1 \times \Gamma_2 \mid p_1(g) = p_2(h) \right\}. \end{align*} $$

We observe now that $\Gamma _1$ and $\Gamma _2$ are finitely presented—i.e., of homotopical type $\text {F}_{2}$ —since

$$ \begin{align*} \ker(p_1) = \left( \begin{smallmatrix} 1 & 0 & * & 0 \\ 0 & 1 & * & 0 \\ 0 & 0 & * & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix} \right) \cong \mathcal{C}_{3} \leq H_1 \, \, \text{ and } \, \, \left( \begin{smallmatrix} 1 & 0 & 0 & 0 \\ 0 & * & 0 & * \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{smallmatrix} \right) \cong \mathbb{A}\text{ff}(R) \leq \operatorname{\textrm{GL}}_2(R) \end{align*} $$

are so. (In particular, $\ker (p_1)$ is of type $\text {F}_{1}$ .) Again, the finite presentability of $\textbf {B}_2^{\circ }(R)$ implies that the abelian group $Q \cong \mathbb {G}_m(R)$ is finitely generated, whence it is of type $\text {F}_{3}$ ; cf. Section 2. Therefore, the fiber product $P \cong H_4$ is finitely presented by the (asymmetric) 1-2-3-Theorem [Reference Bridson, Howie, Miller and Short16, Theorem B].

Entirely analogous arguments for the nontrivial intersections of members of $\mathscr{H}\;(n,R)$ show that all such groups are also finitely presented, which yields the proposition.▪

We now investigate connectivity properties of the complex $CC(\mathscr{H}\;(n,R))$ . The following observation, whose proof we omit, is originally due to Holz [Reference Holz39]. To verify it directly, consider the homotopy equivalences given in [Reference Abels and Holz4, Theorem 1.4]. Alternatively, a generalization has been recently obtained by Brück; see [Reference Brück20, Theorem 3.17 and Corollary 3.18] for a proof.

For our groups, Lemma 4.4 yields the following.

Corollary 4.5 Let $\mathscr{H}_u(n,R)$ denote the family of unipotent radicals $\mathcal {U}_i(R) = H_i(R) \cap \textbf {U}_n(R)$ of members of $\mathscr{H}\;(n,R)$ and consider the following covering of $\textbf {U}_n(R)$ .

$$ \begin{align*} \mathfrak{H}_u(n,R) = \left\{ v \cdot \mathcal{U}(R) \mid v \in \textbf{U}_n(R) \text{ and } \, \mathcal{U}(R) \in \mathscr{H}_u(n,R) \right\}. \end{align*} $$

Then the spaces $CC(\mathscr{H}_u(n,R)) = N(\mathfrak {H}_u(n,R))$ (with respect to $\textbf {U}_n(R)$ ) and $CC(\mathscr{H}\;(n,R))$ (with respect to $\textbf {A}_n(R)$ ) are homotopy equivalent.

Thus, to show that $CC(\mathscr{H}\;(n,R))$ is connected and simply-connected, it suffices to prove that the coset complex $CC(\mathscr{H}_u(n,R))$ of contracting subgroups, with cosets taken in the unipotent radical $\textbf {U}_n(R)$ , is connected and simply-connected. To do so, we take advantage of the algebraic meaning of connectivity properties of coset complexes discovered by Tits [Reference Tits57].

Recall that the colimit $\text {colim}\, F$ of a diagram $F : I \to \text {Grp}$ from a small category I to the category of groups is a group K together with a family of maps $\Psi = \left \{ \psi _O : F(O) \to K \right \}_{O \in \text {Obj}(I)}$ satisfying the following properties.

• • $\psi _P \circ F(f) = \psi _O$ for all $f \in \operatorname {\textrm {Hom}}(O, P)$ .

• • If $(K', \Psi ')$ is another pair also satisfying the conditions above, then there exists a unique group homomorphism $\varphi : K \to K'$ such that $\varphi \circ \psi _O = \psi _O'$ for all $O \in \text {Obj}(I)$ .

In this case, we write $K = \text {colim}\, F$ , omitting the maps $\Psi $ . Now suppose $\mathscr{H}$ is a family of subgroups of a given group. This induces a diagram $F_\mathscr{H} : I_\mathscr{H} \to \text {Grp}$ by defining the category $I_\mathscr{H}$ to be the poset given by members of $\mathscr{H}$ and their pairwise intersections, ordered by inclusion. For example, if $\mathscr{H} = \left \{ A,B \right \}$ with $A, B \leq G$ and $C = A \cap B$ , then $F_\mathscr{H}$ is just the usual diagram $A \hookleftarrow C \hookrightarrow B$ and thus the colimit $\text {colim}\, F_\mathscr{H}$ is simply the push-out (or amalgamated product) $\text {colim}\, F_\mathscr{H} = A \ast _C B$ .

To apply Theorem 4.6 in our context, we will need a bit of commutator calculus. The following identities are well-known; see e.g., [Reference Abels2, Section 2.2].

Lemma 4.7 Let G be a group and let $a,b,c \in G$ . Then

(4.1) $$ \begin{align} [ab,c] = a[b,c]a^{-1}[a,c] \end{align} $$

and

(Hall's identity) $$ \begin{align} [cac^{-1},[b,c]] \cdot [bcb^{-1},[a,b]] \cdot [aba^{-1},[c,a]] = 1. \end{align} $$

We also need a convenient, well-known presentation for $\textbf {U}_n(R)$ . To describe this standard presentation, we need some notation. Fix $T \subseteq R$ a generating set, containing the unit $1$ , for the underlying additive group $\mathbb {G}_a(R)$ of the base ring R. That is, we view R as a quotient of the free abelian group $\bigoplus \limits _{t \in T} \mathbb {Z} t$ .

We fix furthermore $\mathcal {R} \subseteq \bigoplus _{t \in T} \mathbb {Z} t$ a set of additive defining relations of R. In other words, $\mathcal {R}$ is a set of expressions $\left \{ \sum \limits _{\ell } a_{\ell } t_{\ell } \mid a_{\ell } \in \mathbb {Z}, t_{\ell } \in T \right \} \subseteq \bigoplus \limits _{t \in T} \mathbb {Z} t$ (where all but finitely many coefficients $a_{\ell }$ are zero) such that $\mathbb {G}_a(R) \cong \left (\bigoplus \limits _{t \in T} \mathbb {Z} t\right ) / \langle {\mathcal {R}} \rangle _{\mathbb {Z}}$ .

For every pair $t,s\in T$ of additive generators, we choose an expression $m(t,s) = m(s,t) \in \bigoplus _{t \in T} \mathbb {Z} t$ such that the image of $m(s,t)$ in R under the given projection $\bigoplus _{t \in T} \mathbb {Z} t \twoheadrightarrow R$ equals the products $ts$ and $st$ . In case $t = 1$ , we take $m(1,s)$ to be s itself, i.e., $m(1,s) = s = m(s,1)$ .

Lemma 4.8 With the notation above, the group $\textbf {U}_n(R) \leq \textbf {A}_n(R)$ admits a presentation $\textbf {U}_n(R) \cong \langle {\mathcal {Y} \mid \mathcal {S}} \rangle $ with generating set

$$ \begin{align*} \mathcal{Y} = \left\{ e_{ij}(t) \mid t\in T, 1 \leq i < j \leq n \right\}, \end{align*} $$

and a set of defining relations $\mathcal {S}$ given as follows. For all $(i,j)$ with $1 \leq i < j \leq n$ and all pairs $t, s \in T$ ,

(4.2) $$ \begin{align} [e_{ij}(t),e_{kl}(s)] = \begin{cases} \prod\limits_u e_{il}(u)^{a_u}, & \text{ if } j=k;\\ 1, & \text{ if } i \neq l, k \neq j, \end{cases} \end{align} $$

where $m(t,s) = \sum \limits _{u} a_u u \in \bigoplus \limits _{t \in T} \mathbb {Z} t$ is the fixed expression $m(t,s)$ as above for the product $ts = st \in R$ .

For all $(i,j)$ with $1 \leq i < j \leq n$ ,

(4.3) $$ \begin{align} \prod\limits_{\ell} e_{ij}(t_{\ell})^{a_{\ell}} = 1 \text{ for each } \sum\limits_{\ell} a_{\ell} t_{\ell} \in \mathcal{R}. \end{align} $$

The set $\mathcal {S}$ is defined as the set of all relations (4.2) and (4.3) given above.

Lemma 4.8 is far from new, so we omit its proof. The presentation above has been considered many times in the literature, most notably in the case where R is a field and in connection to buildings and amalgams; see e.g., [Reference Steinberg53, Chapter 3], [Reference Tits57], [Reference Tits56, Appendix 2], [Reference Devillers and Mühlherr34], and [Reference Abramenko and Brown5, Chapters 7 and 8]. In general, such presentation is extracted from the commutator relations (2.1) between elementary matrices—recall that, in the Chevalley–Demazure set-up, $\textbf {U}_n$ is a maximal unipotent subscheme (over $\mathbb {Z}$ ) in type $\text {A}_{n-1}$ . The only difference between the presentation we spelled out and other versions typically occurring elsewhere is that we made the ring structure of R more explicit in the relations occurring in $\textbf {U}_n(R)$ . The interested reader is referred e.g., to [Reference Santos Rego48, Section 1.1.2] for a detailed proof of Lemma 4.8.

Using the above results, we shall have the last ingredient for the proof of Theorem 1.2(3) once we establish the following generalization of a result due to Holz [Reference Holz39, Proposition A.3].

Proof The idea is to write down a convenient presentation for $\textbf {U}_n(R)$ which shows that it is the desired colimit. To do so, we first spell out canonical presentations for the members of $\mathscr{H}_u(n,R)$ . For the course of this proof we fix (and follow strictly) the notation of Lemma 4.8. In particular, $T \subseteq R$ will denote an arbitrary, but fixed, additive generating set for $(R,+) = \mathbb {G}_a(R)$ containing $1$ . As in Lemma 4.8, we fix $\mathcal {R}$ a set of additive defining relations of $\mathbb {G}_a(R)$ .

We observe that $\mathcal {U}_3(R)$ and $\mathcal {U}_4(R)$ are abelian, by the commutator relations (2.1). It is also not hard to see that $\mathcal {U}_1(R) \cong \textbf {U}_{n-1}(R) \cong \mathcal {U}_2(R)$ by translating the indices of elementary matrices accordingly. Thus, we have the following presentations.

$$ \begin{align*} \mathcal{U}_1(R) \cong & \langle \left\{ e_{ij}(t) \mid t \in T, \, 1 \leq i < j \leq n-1 \right\} \mid \text{ Relations (4.2) and (4.3)} \\ & \text{for all } i,j \text{ with } 1 \leq i < j \leq n-1, \text{ and all } t,s \in T \rangle. \end{align*} $$

$$ \begin{align*} \mathcal{U}_2(R) \cong & \langle \left\{ e_{ij}(t) \mid t \in T, 2 \leq i < j \leq n \right\} \mid \text{ Relations (4.2) and (4.3)} \\ & \text{for all } i,j \text{ with } 2 \leq i < j \leq n, \text{ and all } t,s \in T \rangle. \end{align*} $$

$$ \begin{align*} \mathcal{U}_3(R) \cong & \langle \left\{ e_{12}(t), e_{n-1,n}(s) \mid t,s \in T \right\} \mid [e_{12}(t), e_{n-1,n}(s)] = 1 \text{ for all } t,s \in T, \\ & \text{and relations (4.3) for all } t,s \in T \text{ and } (i,j) \in \left\{ (1,2), (n-1,n) \right\} \rangle. \end{align*} $$

$$ \begin{align*} \mathcal{U}_4(R) \cong & \langle \left\{ e_{13}(p), \, e_{23}(t), \, e_{24}(s) \mid p,t,s \in T \right\} \, \mid \, [e_{ij}(t), e_{kl}(s)] = 1 \text{ for all } \\ & t, s \in T \mbox { and } (i,j), (k,l) \in \left\{ (1,3),(2,3),(2,4) \right\}, \text{ and} \\ & \text{relations (4.3) for all } t,s \in T \text{ and } (i,j) \in \left\{ (1,3),(2,3),(2,4) \right\} \rangle. \end{align*} $$

The pairwise intersections $\mathcal {U}_i(R) \cap \mathcal {U}_j(R)$ also admit similar presentations by restricting the generators (and corresponding relations) to the indices occurring in both $\mathcal {U}_i(R)$ and $\mathcal {U}_j(R)$ . For instance,

$$ \begin{align*} \mathcal{U}_1(R) \cap \mathcal{U}_2(R) = \left( \begin{smallmatrix} 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & * & \cdots & * & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ & & 0 & 1 & * & 0 \\ & & & 0 & 1 & 0 \\ 0 & \cdots & & \cdots & 0 & 1 \end{smallmatrix} \right) \cong \textbf{U}_{n-2}(R) \end{align*} $$

with presentation

$$ \begin{align*} \mathcal{U}_1(R) \cap \mathcal{U}_2(R) \cong & \langle \left\{ e_{ij}(t) \mid t \in T, \, 2 \leq i < j \leq n-1 \right\} \mid \text{ Relations (4.2) and} \\ & \text{(4.3) for all } i,j \text{ with } 2 \leq i < j \leq n-1, \text{ and all } t,s \in T \rangle. \end{align*} $$

Now consider the group $U_n$ defined as follows. As generating set we take

$$ \begin{align*} \mathcal{X}_n = \left\{ e_{ij}(t), \, e_{kn}(s) \, \mid \, 1 \leq i < j \leq n-1, \, \, 2 \leq k \leq n, \text{ and } t,s \in T \right\}. \end{align*} $$

The set of defining relations $\mathcal {S}_n$ is formed as follows. For all $t,s \in T$ and indices $i,j,k,l$ which are either all in $\left \{ 1,\ldots ,n-1 \right \}$ or all in $\left \{ 2,\ldots ,n \right \}$ , consider the relations

(4.4) $$ \begin{align} [e_{ij}(t),e_{kl}(s)] = \begin{cases} \prod\limits_u e_{il}(u)^{a_u}, & \text{ if } j=k;\\ 1, & \text{ if } i \neq l, k \neq j, \end{cases} \end{align} $$

and

(4.5) $$ \begin{align} [e_{12}(t),e_{n-1,n}(s)] = 1, \end{align} $$

where $m(t,s) = \sum \limits _{u} a_u u \in \bigoplus \limits _{t \in T} \mathbb {Z} t$ is as in Lemma 4.8. For all $t,s \in T$ and pairs $i,j$ which are either all in $\left \{ 1,\ldots ,n-1 \right \}$ or all in $\left \{ 2,\ldots ,n \right \}$ , consider additionally the relations

(4.6) $$ \begin{align} \prod_{\ell=1}^m e_{ij}(t_{\ell})^{a_{\ell}} = 1 \text{ for each } \sum_{\ell=1}^m a_{\ell} t_{\ell} \in \mathcal{R}, \end{align} $$

where $\mathcal {R}$ is the fixed set of additive defining relations of $\mathbb {G}_a(R)$ as in Lemma 4.8. If $n = 4$ we need also consider the relations

(4.7) $$ \begin{align} [e_{13}(t),e_{24}(s)] = 1 \end{align} $$

for all pairs $t,s \in T$ . We take $\mathcal {S}_n$ to be the set of all relations (4.4)–(4.6) (in case $n \geq 5$ ), and $\mathcal {S}_4$ is the set of all relations (4.4) through (4.7) above. We then define $U_n$ by means of the presentation

$$ \begin{align*} U_n = \langle {\mathcal{X}_n \mid \mathcal{S}_n} \rangle. \end{align*} $$

Reading off the presentations for the $\mathcal {U}_i(R)$ and for their pairwise intersections, it follows from von Dyck’s theorem that $\text {colim}\, F_{\mathscr{H}_u(n,R)}$ is isomorphic to the group $U_n$ above.

Thus, to prove the proposition it suffices to show that $\textbf {U}_n(R)$ is isomorphic to $U_n$ . To avoid introducing even more symbols, we proceed as follows. Recall that $\textbf {U}_n(R)$ admits the presentation $\textbf {U}_n(R) \cong \langle {\mathcal {Y} \mid \mathcal {S}} \rangle $ given in Lemma 4.8. Abusing notation and comparing the presentations $U_n = \langle {\mathcal {X}_n \mid \mathcal {S}_n} \rangle $ and $\textbf {U}_n(R) \cong \langle {\mathcal {Y} \mid \mathcal {S}} \rangle $ , it suffices to define in $U_n$ the missing generators $e_{1n}(t)$ (for $t \in T$ ) and also show that all the relations from $\mathcal {S}$ missing from $\mathcal {S}_n$ do hold in $U_n$ . (Inspecting the indices, the missing relations are those involving the commutators $[e_{1j}(t),e_{kn}(s)]$ for $j = 2, \ldots , n$ and $k = 1, \ldots , n-1$ , and $(j,k) \neq (2,n-1)$ and those involving only the new generators $e_{1n}(t)$ .)

For every $t \in T$ , define in $U_n$ the element $e_{1n}(t) = [e_{12}(t), e_{2n}(1)]$ . With this new commutator at hand, the proof will be concluded once we show that the following equalities hold in $U_n$ .

For all $s,t \in T$ and $j,k \in \left \{ 2,\ldots ,n-1 \right \}$ with $j \neq k$ and $(j,k) \neq (2,n-1)$ ,

(4.8) $$ \begin{align} [e_{1j}(t),e_{kn}(s)] = 1. \end{align} $$

For all $t \in T$ and $j \in \left \{ 2,\ldots ,n-1 \right \}$ ,

(4.9) $$ \begin{align} [e_{1j}(t),e_{jn}(1)] = [e_{1j}(1),e_{jn}(t)] = e_{1n}(t). \end{align} $$

For all $t, s \in T$ and $i,j$ with $1 \leq i < j \leq n$ ,

(4.10) $$ \begin{align} [e_{ij}(t),e_{1n}(s)] = 1. \end{align} $$

For all $\sum _{\ell =1}^m a_{\ell } t_{\ell } \in \mathcal {R}$ ,

(4.11) $$ \begin{align} \prod_{\ell=1}^m e_{1n}(t_{\ell})^{a_{\ell}} = 1. \end{align} $$

Relation (4.8) holds: If $n = 4$ , there is nothing to show, since in this case, the only equation to verify is $[e_{13}(t),e_{24}(s)] = 1$ , which holds by (4.7). Assume $n \geq 5$ . We first observe that

(4.12) $$ \begin{align} [e_{1j}(t), e_{n-1,n}(s)] = 1 \end{align} $$

for all $t,s \in T$ and $j \in \left \{ 3,\ldots ,n-2 \right \}$ since

$$ \begin{align*} e_{1j}(t) e_{n-1,n}(s) & \overset{\text{(4.4)}}{=} e_{12}(t) e_{2j}(1) e_{12}(t)^{-1} e_{2j}(1)^{-1} e_{n-1,n}(s) \\ & \overset{\text{(4.4)}}{=} e_{12}(t) e_{2j}(1) e_{12}(t)^{-1} e_{n-1,n}(s) e_{2j}(1)^{-1} \\ & \overset{\text{(4.5)}}{=} e_{12}(t) e_{2j}(1) e_{n-1,n}(s) e_{12}(t)^{-1} e_{2j}(1)^{-1} \\ & \overset{\text{(4.4)}}{=} e_{12}(t) e_{n-1,n}(s) e_{2j}(1) e_{12}(t)^{-1} e_{2j}(1)^{-1} \\ & \overset{\text{(4.5)}}{=} e_{n-1,n}(s) e_{12}(t) e_{2j}(1) e_{12}(t)^{-1} e_{2j}(1)^{-1} \\ & \overset{\text{(4.4)}}{=} e_{n-1,n}(s) e_{1j}(t). \end{align*} $$

Proceeding similarly, we conclude that

(4.13) $$ \begin{align} [e_{12}(t), e_{kn}(s)] = 1 \end{align} $$

for all $t, s \in T$ and $k \in \left \{ 3,\ldots ,n-2 \right \}$ . Now suppose $j < k$ . Then

$$ \begin{align*} [e_{1j}(t),e_{kn}(s)] \overset{\text{(4.4)}}{=} [e_{1j}(t), [e_{k,n-1}(s),e_{n-1,n}(1)]] = 1 \end{align*} $$

because $e_{1j}(t)$ commutes with $e_{n-1,n}(1)$ , by (4.12), and with $e_{k,n-1}(s)$ , by (4.4). Analogously, if $j> k$ , then

$$ \begin{align*} [e_{1j}(t),e_{kn}(s)] \overset{\text{(4.4)}}{=} [[e_{12}(t),e_{2j}(1)],e_{kn}(s)] = 1 \end{align*} $$

by (4.4) and (4.13). Thus, the relations (4.8) hold in $U_n$ .

Relation (4.9) holds: To check (4.9) we need Lemma 4.7. First,

$$ \begin{align*} [e_{12}(t),e_{2n}(1)] \overset{\text{(4.4)}}{=} [e_{12}(t),[e_{23}(1),e_{3n}(1)]]. \end{align*} $$

Setting $a = e_{12}(t)$ , $b = e_{23}(1)$ , and $c = e_{3n}(1)$ , Hall's identity yields

$$ \begin{align*} 1 & = [cac^{-1},[b,c]] \cdot [bcb^{-1},[a,b]] \cdot [aba^{-1},[c,a]] \\ & \overset{\text{(4.4)}}{=} [e_{12}(t),e_{2n}(1)] \cdot [e_{23}(1) e_{3n}(1) e_{23}(1)^{-1}, e_{13}(t)] \\ & \overset{\text{(4.4)}}{=} [e_{12}(t),e_{2n}(1)] \cdot [e_{2n}(1) e_{3n}(1), e_{13}(t)] \\ & \overset{\text{(4.1)}}{=} [e_{12}(t),e_{2n}(1)] \cdot e_{2n}(1) \cdot [e_{3n}(1),e_{13}(t)] \cdot e_{2n}(1)^{-1} \cdot [e_{2n}(1),e_{13}(t)] \\ & \overset{\text{(4.4)&(4.8)}}{=} [e_{12}(t),e_{2n}(1)] \cdot [e_{3n}(1),e_{13}(t)], \end{align*} $$

that is, $e_{1n}(t) = [e_{13}(t),e_{3n}(1)]$ . On the other hand,

$$ \begin{align*} [e_{12}(1),e_{2n}(t)] \overset{\text{(4.4)}}{=} [e_{12}(1),[e_{23}(t),e_{3n}(1)]]. \end{align*} $$

Setting $a = e_{12}(1)$ , $b = e_{23}(t)$ , and $c = e_{3n}(1)$ , Hall's identity and (4.4) yield

$$ \begin{align*} 1 & {=} [e_{12}(1),e_{2n}(t)] \cdot [e_{23}(t) e_{3n}(1) e_{23}(1)^{-1}, e_{13}(t)] \\ & \overset{\text{(4.4)}}{=} [e_{12}(1),e_{2n}(t)] \cdot [e_{2n}(t) e_{3n}(1), e_{13}(t)] \\ & \overset{\text{(4.1)}}{=} [e_{12}(1),e_{2n}(t)] \cdot e_{2n}(t) \cdot [e_{3n}(1),e_{13}(t)] \cdot e_{2n}(t)^{-1} \cdot [e_{2n}(t),e_{13}(t)] \\ & \overset{\text{(4.4)&(4.8)}}{=} [e_{12}(1),e_{2n}(t)] \cdot [e_{3n}(1),e_{13}(t)]. \end{align*} $$

The last product above equals $[e_{12}(1),e_{2n}(t)] e_{1n}(t)^{-1}$ by the previous computations. We have thus proved that

$$ \begin{align*} e_{1n}(t) \overset{\text{Def.}}{=} [e_{12}(t),e_{2n}(1)] = [e_{12}(1),e_{2n}(t)] = [e_{13}(t),e_{3n}(1)]. \end{align*} $$

Since $[e_{12}(1),e_{2n}(t)]$ also equals $[e_{12}(1),[e_{23}(1),e_{3n}(t)]]$ , again by (4.4), computations similar to the above also yield $[e_{13}(t),e_{3n}(1)] = [e_{13}(1),e_{3n}(t)]$ . Entirely analogous arguments show that

$$ \begin{align*} [e_{1j}(t),e_{jn}(1)] = [e_{1j}(1),e_{jn}(t)] = e_{1n}(t) \end{align*} $$

for all $j \in \left \{ 2, \ldots , n-1 \right \}$ .

Relations (4.10) hold: We now prove that the subgroup $Z :=~\langle {\,\left \{ e_{1n}(t) \mid t \in T \right \}\,} \rangle \leq U_n$ is central. Let $t, s \in T$ and let $i,j$ be such that $1 \leq i < j \leq n$ . We want to show that $e_{1n}(t)$ and $e_{ij}(s)$ commute in $U_n$ . To begin with,

$$ \begin{align*} [e_{1n}(t),e_{1n}(s)] \overset{\text{(4.9)}}{=} [[e_{12}(t),e_{2n}(1)],[e_{13}(s),e_{3n}(1)]] \overset{\text{(4.4)}}{=} 1, \end{align*} $$

i.e., Z is abelian. If $i = 1$ and $j \neq n$ , then $j \geq 2$ and we can pick $k \in ~\left \{ 2, \ldots , n-1 \right \}$ such that $k \neq j$ because $n \geq 4$ , yielding

$$ \begin{align*} e_{1n}(t) e_{1j}(s) \overset{\text{(4.9)}}{=} [e_{1k}(t), e_{kn}(1) ] e_{1j}(s) & \overset{\text{(4.8)&(4.4)}}{=} e_{1j}(s) [e_{1k}(t), e_{kn}(1)] \\ & \overset{{\phantom{111}(4.9)\phantom{x11}}}{=} e_{1j}(s) e_{1n}(t). \end{align*} $$

Similarly, if $j = n$ and $i \neq 1$ , choose $k \in \left \{ 2, \ldots , n-1 \right \}$ such that $k \neq i$ . We obtain

$$ \begin{align*} [e_{1n}(t),e_{in}(s)] \overset{\text{(4.9)}}{=} [[e_{1k}(t),e_{kn}(1)],e_{in}(s)] \overset{\text{(4.4)&(4.8)}}{=} 1. \end{align*} $$

It remains to prove $[e_{1n}(t), e_{ij}(s)] = 1$ for $1 < i < j < n$ . In this case,

$$ \begin{align*} e_{1n}(t) e_{ij}(s) & \overset{{\phantom{x11}(4.9)\phantom{x11}}}{=} e_{1i}(1) e_{in}(t) e_{1i}(1)^{-1} e_{in}(t)^{-1} e_{ij}(s) \\ & \overset{{\phantom{x11}(4.4)\phantom{x11}}}{=} e_{1i}(1) e_{in}(t) e_{1i}(1)^{-1} e_{ij}(s) e_{in}(t)^{-1} \\ & \overset{\text{(4.6)&(4.4)}}{=} e_{1i}(1) e_{in}(t) e_{1j}(s)^{-1} e_{ij}(s) e_{1i}(1)^{-1} e_{in}(t)^{-1} \\ & \overset{\text{(4.8)&(4.4)}}{=} e_{1j}(s)^{-1} e_{1i}(1) e_{in}(t) e_{ij}(s) e_{1i}(1)^{-1} e_{in}(t)^{-1} \\ & \overset{{\phantom{x11}(4.4)\phantom{x11}}}{=} e_{1j}(s)^{-1} e_{1i}(1) e_{ij}(s) e_{in}(t) e_{1i}(1)^{-1} e_{in}(t)^{-1} \\ & \overset{\text{(4.6)&(4.4)}}{=} e_{1j}(s)^{-1} e_{1j}(s) e_{ij}(s) e_{1i}(1) e_{in}(t) e_{1i}(1)^{-1} e_{in}(t)^{-1} \\ & \overset{{\phantom{x11}(4.9)\phantom{x11}}}{=} e_{ij}(s) e_{1n}(t). \end{align*} $$

Thus, relations (4.10) hold for all $t, s \in T$ and $i,j$ with $1 \leq i < j \leq n$ .

Relations (4.11) hold: Given any pair $t,s \in T$ ,

(4.14) $$ \begin{align} \nonumber [e_{12}(t) e_{12}(s), e_{2n}(1)] & \overset{{\phantom{x11}(4.1)\phantom{x11}}}{=} e_{12}(t)[e_{12}(s), e_{2n}(1)] e_{12}(t)^{-1} [e_{12}(t), e_{2n}(1)] \\ \nonumber & \overset{\text{(4.9)&(4.10)}}{=} [e_{12}(s), e_{2n}(1)] [e_{12}(t), e_{2n}(1)] \\ & \overset{\text{(4.9)&(4.10)}}{=} [e_{12}(t), e_{2n}(1)] [e_{12}(s), e_{2n}(1)]. \end{align} $$

Now let $\sum \limits _{\ell = 1}^m a_{\ell } t_{\ell } \in \mathcal {R}$ be an additive defining relation in R. (Recall that $t_{\ell } \in T$ and $a_{\ell } \in \mathbb {Z}$ as in Lemma 4.8.) Induction on $\sum \limits _{\ell = 1}^{m} |a_{\ell }|$ and (4.14) yield

$$ \begin{align*} \prod\limits_{\ell = 1}^{m} e_{1n}(t_{\ell})^{a_{\ell}} \overset{\text{Def.}}{=} \prod\limits_{\ell = 1}^{m} \left([e_{12}(t_{\ell}), e_{2n}(1)]\right)^{a_{\ell}} \overset{\text{(4.14)}}{=} \left[ \prod\limits_{\ell = 1}^{m} e_{12}(t_{\ell})^{a_{\ell}}, e_{2n}(1) \right] \overset{\text{(4.4)}}{=} 1. \end{align*} $$

Since the relations (4.8)–(4.11) missing from the presentation for $\textbf {U}_n(R)$ from Lemma 4.8 also hold in $U_n$ , it follows that $U_n \cong \textbf {U}_n(R)$ , as claimed.▪

4.2 Proof of Theorem 1.2

If $\phi (\textbf {A}_n(R))> 0$ for some $n \geq 3$ , then $\textbf {A}_n(R)$ and its retract $\mathbb {A}\text {ff}(R) \cong { \left ( \begin {smallmatrix} * & * \\ 0 & 1 \end {smallmatrix} \right ) } \leq \operatorname {\textrm {GL}}_2(R)$ are finitely generated. Thus, the abelian group $\mathbb {G}_m(R)$ is finitely generated and $\mathbb {G}_a(R)$ is finitely generated as a $\mathbb {Z}[\mathbb {G}_m(R)]$ -module, which shows that R is finitely generated as a ring. This deals with the very first claim of Theorem 1.2 (except possibly when $n = 2$ ).

Now, if $\phi (\textbf {A}_2(R))> 0$ , then $\textbf {A}_2(R) \cong \mathbb {G}_a(R)$ is finitely generated as a $\mathbb {Z}$ -module. This implies, for every $n \geq 2$ , that the unipotent radical $\textbf {U}_n(R)$ of $\textbf {A}_n(R)$ is a finitely generated nilpotent group and thus has $\phi (\textbf {U}_n(R)) = \infty $ . (The equality follows e.g., from Lemma 2.1 by induction on the nilpotency class because all terms of the lower central series of a finitely generated nilpotent group are themselves finitely generated.) Moreover, $\textbf {A}_2(R) \cong \mathbb {G}_a(R)$ being finitely generated also implies that the group of units $\mathbb {G}_m(R)$ is finitely generated by Samuel’s generalization of Dirichlet’s unit theorem [Reference Samuel49, Section 4.7]. Thus, $\textbf {D}_n(R)\leq \operatorname {\textrm {GL}}_n(R)$ and $\textbf {T}_n(R) \leq \textbf {A}_n(R)$ also have $\phi (\textbf {T}_n(R)) = \phi (\textbf {D}_n(R)) = \infty $ . Since $\textbf {A}_n(R) = \textbf {U}_n(R) \rtimes \textbf {T}_n(R)$ and $\textbf {B}_2(R) = \textbf {U}_2(R) \rtimes \textbf {D}_2(R)$ , it follows from Lemmata 2.1 and 3.3 that $\phi (\textbf {A}_n(R)) = \phi (\textbf {B}_2^{\circ }(R)) = \infty $ .

Assume from now on that R is not finitely generated as a $\mathbb {Z}$ -module. For $n \geq 3$ , we first observe that $\phi (\textbf {A}_n(R)) \geq 1$ whenever $\phi (\textbf {B}_2^{\circ }(R)) \geq 1$ . Indeed, assume the latter, i.e., $\textbf {B}_2^{\circ }(R)$ is finitely generated. Then, for every $i \in \left \{ 2, \ldots , n-2 \right \}$ , the subgroups

$$ \begin{align*} \textbf{E}_{i,i+1}(R) \rtimes (D_i(R) \times D_{i+1}(R)) \cong \textbf{B}_2(R), \end{align*} $$

$$ \begin{align*} \textbf{E}_{12}(R) \rtimes D_2(R) \cong \mathbb{A}\text{ff}_-(R) \cong {\left( \begin{smallmatrix} 1 & * \\ 0 & * \end{smallmatrix} \right) } \leq \operatorname{\textrm{GL}}_2(R), \, \text{ and} \end{align*} $$

$$ \begin{align*} \textbf{E}_{n-1,n}(R) \rtimes D_{n-1}(R) \cong \mathbb{A}\text{ff}(R) \cong {\left( \begin{smallmatrix} * & * \\ 0 & 1 \end{smallmatrix} \right) } \leq \operatorname{\textrm{GL}}_2(R) \end{align*} $$

are also finitely generated by Lemma 3.2. By relations (2.1) and (2.2), the subgroups above generate all of $\textbf {A}_n(R)$ , whence $\phi (\textbf {A}_n(R)) \geq 1$ . Secondly, it is straightforward that $\textbf {A}_n(R)$ —still for $n \geq 3$ —retracts e.g., onto $\mathbb {A}\text {ff}(R)$ , which implies $\phi (\textbf {A}_n(R)) \leq \phi (\textbf {B}_2^{\circ }(R))$ by Theorem 1.1.

From the observations above, the proof of part (2) will be concluded once we show that $\textbf {A}_3(R)$ can never be finitely presented, i.e., $\phi (\textbf {A}_3(R)) \leq 1$ . To see this, in case $\mathbb {G}_m(R)$ is itself finite or is infinitely generated, then $\phi (\textbf {A}_3(R)) = \phi (\textbf {U}_3(R)) = 0$ . In case $\mathbb {G}_m(R)$ has torsion-free rank at least one and if $\textbf {A}_3(R)$ were finitely presented, then its metabelian quotient $\textbf {A}_3(R)/Z(\textbf {A}_3(R)) = \textbf {A}_3(R) / \textbf {E}_{13}(R)$ would also be finitely presented by [Reference Bieri, de Cornulier, Guyot and Strebel13, Corollary 5.6]. But the complement of the $\Sigma $ -invariant [Reference Bieri, de Cornulier, Guyot and Strebel13] of the $\mathbb {Z}[\textbf {T}_1(R)]$ -module $\textbf {U}_3(R) / \textbf {E}_{13}(R) \cong \textbf {E}_{12}(R) \oplus \textbf {E}_{23}(R)$ is readily seen to contain antipodal points, which for us means that $\textbf {U}_3(R) / \textbf {E}_{13}(R)$ is not tame as a $\mathbb {Z}[\textbf {T}_1(R)]$ -module. This would contradict [Reference Bieri, de Cornulier, Guyot and Strebel13, Theorem 5.1] and we are done with part (2).

Turning to part (3) after the previous remarks, it remains to check for $n\geq 4$ that $\phi (\textbf {A}_n(R)) \geq 2$ when $\phi (\textbf {B}_2^{\circ }(R)) \geq 2$ . Suppose the latter holds, i.e., $\textbf {B}_2^{\circ }(R)$ is finitely presented. By Lemma 4.2, the group $\textbf {A}_n(R)$ for $n \geq 4$ acts cocompactly and by cell-permuting homeomorphisms on the simplicial complex $CC(\mathscr{H}\;(n,R))$ . Since $\phi (\textbf {B}_2^{\circ }(R)) \geq 2$ , we know from Lemma 4.2 and Proposition 4.3 that the stabilizer in $\textbf {A}_n(R)$ of any cell of $CC(\mathscr{H}\;(n,R))$ is finitely presented. Since $CC(\mathscr{H}\;(n,R))$ (for $n \geq 4$ ) is connected and simply-connected by Proposition 4.9 and Theorem 4.6, it follows from Theorem 4.1 that $\textbf {A}_n(R)$ is finitely presented. That is, $\phi (\textbf {A}_n(R)) \geq 2$ , as required.▪

4.3 Conjecture 1.7 in the arithmetic set-up

We close the paper by spelling out a proof of the following special case of Conjecture 1.7. Though it has not appeared in this general form in the literature before, we claim no originality—it is a simple combination of famous results mentioned in the introduction.

Proposition 4.10 Let $\mathcal {O}_S$ be a Dedekind domain of arithmetic type. If either $ \operatorname {\textrm {char}}(\mathcal {O}_S) = 0$ or if $ \operatorname {\textrm {char}}(\mathcal {O}_S)> 0$ and S has at most three elements, then the S-arithmetic Abels’ groups $\textbf {A}_n(\mathcal {O}_S)$ satisfy Conjecture 1.7. That is,

$$ \begin{align*} \phi(\textbf{A}_n(\mathcal{O}_S)) = \min\left\{ n-2, \phi(\textbf{B}_2^{\circ}(\mathcal{O}_S)) \right\} \text{ for all } n \geq 2. \end{align*} $$

Proof If $ \operatorname {\textrm {char}}(\mathcal {O}_S) = 0$ , the Kneser–Tiemeyer local-global principle [Reference Tiemeyer55, Theorem. 3.1] allows us to assume that S contains a single nonarchimedean place. Also, the equality $\phi (\textbf {B}_2^{\circ }(\mathcal {O}_S)) = \infty $ holds by [Reference Tiemeyer55, Corollary 4.5]. By restriction of scalars (see e.g., [Reference Margulis43, Lemma 3.1.4]), it suffices to consider the case where $ \operatorname {\textrm {Frac}}(\mathcal {O}_S) = \mathbb {Q}$ . In this set-up, $\mathcal {O}_S$ is of the form $\mathcal {O}_S = \mathbb {Z}[1/p]$ for some prime number $p \in \mathbb {N}$ . Here, $\textbf {A}_n(\mathbb {Z}[1/p])$ has $\phi (\textbf {A}_n(\mathbb {Z}[1/p])) < n-1$ , for otherwise it would be of homological type $\text {FP}_{n-1}$ and thus of type $\text {FP}_{\infty }$ by [Reference Bieri12, Proposition]. In particular, its center $Z(\textbf {A}_n(\mathbb {Z}[1/p]))$ would be finitely generated by [Reference Bieri12, Corollary 2]. However, $Z(\textbf {A}_n(\mathbb {Z}[1/p]))$ is the elementary subgroup $\textbf {E}_{1n}(\mathbb {Z}[1/p]) \cong \mathbb {G}_a(\mathbb {Z}[1/p])$ , which is not finitely generated, yielding a contradiction. Since $\phi (\textbf {A}_n(\mathbb {Z}[1/p])) \geq n-2$ by [Reference Abels and Brown3, Theorem B] and Brown’s criteria (Theorem 4.1 and [Reference Brown18, Proposition 1.1]), we obtain

$$ \begin{align*} \phi(\textbf{A}_n(\mathcal{O}_S)) = n - 2 = \min\{ n-2, \infty\} = \min\{ n-2, \phi(\textbf{B}_2^{\circ}(\mathcal{O}_S))\}. \end{align*} $$

In case $ \operatorname {\textrm {char}}(\mathcal {O}_S)> 0$ , we have $\phi (\textbf {B}_2^{\circ }(\mathcal {O}_S)) = |S|-1$ by Theorem 1.3. Thus, if S has at most three elements, it follows from Theorem 1.2 that

$$ \begin{align*} \phi(\textbf{A}_n(\mathcal{O}_S)) = \min \{ 0, 1, 2, n-2\} = \min \{ n-2, \phi(\textbf{B}_2^{\circ}(\mathcal{O}_S))\}. \end{align*} $$

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