Notation 2.1. Let ${\mathbb {N}}^*$ denote the set of strings with natural numbers as entries. We use letters $\sigma , \tau , \rho ,$ etc., for elements of ${\mathbb {N}}^*$ . The set ${\mathbb {N}}^*$ can be seen as a directed tree: the empty string is the root, and the successor relation is given by appending a number at the end of a string. One writes $\sigma \preceq \tau $ to denote that $\sigma $ is an initial segment of $\tau $ , and $\sigma \prec \tau $ to denote that $\sigma $ is a proper initial segment. For $k \le |\tau |$ , by $\tau \! \upharpoonright _{k}$ one denotes the initial segment of $\tau $ that has length k. We can also identify finite strings of length n+ $1$ with partial functions ${\mathbb {N}} \rightarrow {\mathbb {N}}$ having finite support $\{0, \ldots , n\}$ . We then write $\tau _i$ instead of $\tau (i)$ . By $\max (\tau )$ we denote $\max \{\tau _i \colon \, i \le n\}$ . Let $h \colon {\mathbb {N}}^* \to {\mathbb {N}}$ be the canonical encoding given by $h(w)= \prod _{i< |w|} p_i^{w_i+1}$ , where $w = (w_0, \ldots , w_{|w|-1})$ , and $p_i$ is the i-th prime number. By convention $h()=1$ .

Proof. (i) The case of union is trivial. For the intersection operation, it suffices to consider the case that u and w are singletons. For strings $\alpha , \beta \in T$ , one has $[\alpha ]_T \cap [\beta ]_T = \emptyset $ if $\alpha , \beta $ are incompatible, and otherwise $[\alpha ]_T \cap [\beta ]_T = [\gamma ]_T$ where $\gamma $ is the longest common initial segment of $\alpha , \beta $ .

(ii) Let H be a computable binary function as in Definition 2.4. It suffices to consider the case that u is a singleton. Suppose that $\alpha \in T^{[\ge k]}$ . The algorithm to decide whether $[\alpha ]_T \subseteq \mathcal {K}_w$ is as follows. Let N be the maximum length of a string in w. Answer “yes” if for each $\beta \succeq \alpha $ of length N such that $\beta (i) \le H(\alpha \! \upharpoonright _{k}, k)$ for each $i < N$ , there is $\gamma \in w$ such that $\gamma \preceq \beta $ . Otherwise, answer “no.”

Proof. “ $\Rightarrow $ ”: Suppose that G is computably t.d.l.c. as a closed subgroup of $S_\infty $ . To save on notation, we may assume that G itself is a closed subgroup of $S_\infty $ showing this. We will obtain a Haar computable copy of its meet groupoid $\mathcal {W}(G)$ .

Recall from Definition 2.7 that $E=E_{{Tree}(G) }\subseteq {\mathbb {N}}$ is the decidable set of minimal code numbers for compact open subsets of G. For $u \in E$ we can decide whether $\mathcal {K}_u$ is a subgroup: this is the case precisely when $\mathcal {K}_{M(u,u)}= \mathcal {K}_u$ and $\mathcal {K}_{I(u)} = \mathcal {K}_u$ , where the computable functions $I,M$ were defined in Lemma 3.5. We can also decide whether $B=\mathcal {K}_v$ is a coset: this is the case precisely when $B B^{-1}$ is a subgroup U (in which case B is its right coset). Equivalently, $\mathcal {K}_{M(v,I(v))}$ is a subgroup, which is a decidable condition.

The domain D of the computable copy of $\mathcal {W}(G)$ is the subset of $ E$ consisting of the code numbers for cosets. For cosets $A= \mathcal {K}_u, B= \mathcal {K}_v \in \mathcal {W}(G)$ recall that $A \cdot B$ is defined iff A is a left coset of a subgroup V such that B is a right coset of V. Equivalently, $\mathcal {K}_{M(I(u), u)}= \mathcal {K}_{M(v,I(v))}$ , which is a decidable condition. So the groupoid operations and meet operation are computable.

It remains to show that, $u,v\in D$ coding subgroups $U,V$ , one can compute $|U : U \cap V|$ . To do so, one enumerates (code numbers in D of) left cosets of $U \cap V$ contained in U, until their union equals U. (There is an algorithm to decide the latter condition by Lemma 2.6(i).) Then one outputs the number of cosets found.

“ $\Leftarrow $ ”: The basic idea is that the left translation action of G on the given Haar computable presentation $\mathcal {W}$ of its meet groupoid, namely $\langle g, A \rangle \mapsto gA$ for $g \in G, A \in \mathcal {W}$ , yields a computable presentation of G via a closed subgroup of $S_\infty $ . The main work is to show that the group $\widetilde G$ of permutations of $\mathcal {W}$ so obtained can be described by a c.l.c. tree (Definition 2.4) whose paths can be viewed as permutations.

To carry this out, we introduce an operator $\mathcal {G}_{\text {comp}} $ from meet groupoids to permutation groups that will be needed frequently later on. In case that $\mathcal {W}$ is a copy of the meet groupoid of G, the operator produces the permutation group given by this left action of G. However, the operator can be defined based on the meet groupoid alone, and thus can be seen as a dual of the operator $G \to \mathcal {W}(G)$ obtained in the first part of the proof.

We emphasise that the elements of $S_\infty $ are not actually permutations, but paths p on $ {Tree}(S_\infty ) $ encoding pairs consisting of a permutation and its inverse. Nonetheless, if $A \in \mathcal {W}$ is denoted by x, below we often suggestively write $p(A)$ for the element of $\mathcal {W}$ denoted by $p(2x)$ .

The formal conditions expressing (a) and (b) state that:

• – for all $x,y \in D$ , we have $p(2 (x \cap y))= p(2x) \cap p(2y)$ ;

• – if $x \cdot y$ is defined then so is $p(2x) \cdot y$ , and $p(2x) \cdot y= p(2(x\cdot y))$ .

Note that such p can be identified with the automorphisms of the structure obtained from $\mathcal {W}$ which, instead of composition, for each B has a partial unary operation $A \mapsto A \cdot B$ .

Suppose now that $\mathcal {W}$ is as in Definition 4.6. Then there is an isomorphism of meet groupoids $ \mathcal {W} \to \mathcal {W}(G)$ , which below we will use to identify $\mathcal {W}$ and $\mathcal {W}(G)$ . We can make the assumptions that $0\in D$ , and that $0$ denotes a subgroup U in $\mathcal {W}$ , without affecting the uniformity statement of the theorem: otherwise we can search $\mathcal {W}$ for the least n such that n is a subgroup, and then work with a new copy of $\mathcal {W}(G)$ where the roles of $0$ and n are swapped.

We note that for each subgroup $U\in \mathcal {W}(G)$ , the set $B=p(U)$ is a left coset of U since $B\cdot U = p(U)\cdot U= p(U\cdot U)= B$ . Define a group homomorphism $\Phi \colon G \to \widetilde G$ by letting $\Phi (g)$ be the element of $S_\infty $ corresponding to the left translation by g, i.e., $A \mapsto gA$ where $A \in \mathcal {W}(G)$ . (See the comment after Definition 5.2.) Note that $\Phi $ is injective because the compact open subgroups form a neighbourhood basis of $1$ : if $g \neq 1$ then $g \not \in U$ for some compact open subgroup U, so that $\Phi (g)(U) \neq U$ .

To show that $\Phi $ is onto, let $p \in \widetilde G$ . Clearly the subgroups form a filter in $\mathcal {W}(G)$ . Using (a) this implies that $\{p(U)\colon U \in \mathcal {W}(G) \text { is a subgroup}\}$ is also a filter on $\mathcal {W}(G)$ . Since this filter contains a compact set, there is an element g in its intersection. Then $\Phi (g)= p$ : since the set $B=p(U)$ is a left coset of U, it equals $gU$ . So, if A is a right coset of U, then $p(A)= p(U\cdot A)= B\cdot A = gA$ .

To show that $\Phi $ is continuous at $1$ (and hence continuous), note that a basis of neighbourhoods of the identity in $\widetilde G$ is given by the open sets

$$ \begin{align*} \{ p\in \widetilde G \colon \forall i \le n \, [p(A_i) = A_i]\}, \end{align*} $$

where $A_1, \ldots , A_n \in \mathcal {W}(G)$ . Given such a set, suppose $A_i$ is a right coset of $U_i$ , and let $U = \bigcap U_i$ . If $g \in U$ then $gA_i = A_i$ for each i. By the open mapping theorem for Hausdorff groups Remark 1.2 $\Phi $ is open. This verifies the claim.

To complete the proof of the implication “ $\Leftarrow $ ” of the theorem, we now show that ${Tree}(\widetilde G) $ is c.l.c. as in Definition 2.4, so that $\widetilde G$ is a computable presentation of G via a closed subgroup of $S_\infty $ as required. The following claim will be used to show that ${Tree}(\widetilde G) $ is computable, using the assumption that $\mathcal {W}$ is a computable meet groupoid. As usual by $A,B$ we denote compact open cosets of G, identified here with elements of $D \subseteq {\mathbb {N}}$ , the domain of $\mathcal {W}$ .

“ $\Leftarrow $ ”: Let g be an element of the intersection. Then $gA = B\cdot A^{-1} \cdot A = B=\alpha (A)$ for each $A \in \mathrm {dom} (\alpha )$ .

“ $\Rightarrow $ ”: Suppose $p\in \widetilde G $ extends $\alpha $ . By Claim 5.3, there is $g\in G$ such that $p = \Phi (g)$ . Then $gA = p(A) = B$ for each $A,B$ such that $\alpha (A)= B$ . Such $A,B$ are right cosets of the same subgroup. Hence $B\cdot A^{-1}$ is defined, and clearly g is in the intersection. This establishes the claim.

Recall from (1) in Section 3.1 that a string $ \sigma \oplus \tau \in {Tree}(S_\infty ) $ gives rise to a finite injection $\alpha _{ \sigma \oplus \tau }$ , defined by $\alpha _{ \sigma \oplus \tau }(r)=s $ iff $\sigma (r)= s \, \lor \, \tau (s)=r$ . So

$$\begin{align*}S = \{\sigma \oplus \tau \colon \, \alpha_{ \sigma \oplus \tau} \text{ can be extended to some } p \in \widetilde G\}\end{align*}$$

is a subtree of ${Tree}(S_\infty ) $ with no leaves that is computable because the condition in Claim 5.4 is decidable. Clearly $\widetilde G= [S]$ , and hence $S={Tree}(\widetilde G) $ .

Claim 5.6 below will verify that $S $ is a c.l.c. tree in the sense of Definition 2.4 by providing a computable bound H. The following lemma does the main work, and will also be used later on, such as in the proof of Proposition 8.2 below. Informally, it says that given some subgroup $U \in \mathcal {W}$ , if one declares that $p\in \widetilde G$ has a value $L\in \mathcal {W}$ at U, then one can compute for any $F \in \mathcal {W}$ the finite set of possible values of p at F.

To see this, first one computes $V= F\cdot F^{-1} $ , so that F is a right coset of the subgroup V. Next one computes $k=|U \colon U \cap V|$ , the number of left cosets of $U \cap V$ in U. Note that $\mathcal {L}_0= \{ p(U \cap V) \colon p \in [S] \, \land \, p(U) = L \}$ is the set of left cosets of $U \cap V$ contained in L. Clearly this set has size k. By searching $\mathcal {W}$ until all of its elements have appeared, one can compute a strong index for this set. Next one computes a strong index for the set $\mathcal {L}_1$ of left cosets E of V such that $C \subseteq E$ for some $C \in \mathcal {L}_0$ (this uses that, given C, one can compute E). Finally one outputs a strong index for the set $\{ E F \colon \, E \in \mathcal {L}_1\}$ , which equals $\mathcal {L}$ . This proves the lemma.

For the following recall that the computable set $D\subseteq {\mathbb {N}}$ is the domain of $\mathcal {W}$ , and that $0 \in D$ denotes a subgroup U. If $a,b \in D$ denote left cosets $L_0,L_1$ of U, then the string $(a,b) $ is extended by a path $p\in [S]$ iff $p(U)= L_0$ and $p^{-1}(U)= L_1$ .

To see this, let $v=\lfloor i/2 \rfloor $ . If $v \not \in D$ , let $H(\sigma , i)= i$ . Otherwise, let F be the coset denoted by v. If $i= 2v$ , let L be the coset denoted by a. If $i= 2v+1$ , let L be the coset denoted by b. Applying Lemma 5.5 to $U, L, F$ , one can compute $H(\sigma , i) $ as the greatest number denoting an element of $\{ p(F) \colon \, p \in [S] \, \land \, p(U) = L \}$ .

Notation 6.4. Suppose that a function P is as in Definition 6.1 above, and suppose that for each $\sigma \in T$ , the value $P(\sigma , n)$ is only defined for finitely many n. We write $P(\sigma )= \rho $ if $\rho \in {\mathbb {N}}^*$ is the string of maximal length such that $P(\sigma , n)= \rho (n)$ for each $n < |\rho |$ .

Fact 6.6. The group operations of $S_\infty $ are computable.

Proof. We define partial computable monotonic functions $P_1$ and $P_2$ on ${Tree}(S_\infty ) $ and ${Tree}(S_\infty ) ^2$ , respectively, according to Remark 6.5. For the inverse, let $P_1(\sigma _0 \oplus \sigma _1) = \sigma _1 \oplus \sigma _0$ , where $|\sigma _0|=|\sigma _1|$ . For the binary group operation, given strings $\tau =\tau _0 \oplus \tau _1$ and $\sigma =\sigma _0 \oplus \sigma _1$ of the same, even length, let t be greatest such that for each $r<t$ , $ \rho _0(r):=\tau _0(\sigma _0(r))$ and $\rho _1(r):=\sigma _1(\tau _1(s))$ are defined. Let $P_2(\tau , \sigma ) = \rho _0\oplus \rho _1$ , a string of length $2t$ .

Proof. (i) Given $i \in {\mathbb {N}}$ , consider the subtree of K consisting of the strings $\rho $ such that $P(\rho , i)$ is undefined. If this subtree is infinite, then there is a path $f \in [K]$ such that $\Phi (f)(i)$ is undefined, contradiction. Thus for each n a possible value $g(n)$ as above exists. Using the hypotheses on K, given n one can search for the least such value and output it as $g(n)$ .

(ii) To show that $\Phi ^{-1}$ is computable, we define a partial computable function $Q \colon [S] \times {\mathbb {N}} \to {\mathbb {N}}$ . Clearly $[S]$ is compact and $\Phi ^{-1}$ is continuous. Hence there is a computable function h with $h(t) \ge t$ for each t as follows: given $s \in {\mathbb {N}}$ , for each $\rho \in K$ of length $g(h(s))$ , the values $P(\rho ,i)$ for $i< h(s)$ together determine $\rho \! \upharpoonright _{s}$ . For a string $\beta \in S$ of length $h(s)$ and $k<s$ , define $Q(\beta ,k)= \rho (k)$ where $\rho \in K$ is a string of length $g(h(s))$ with $P(\rho ,i) = \beta (i)$ for each $i< h(s)$ . It is easy to verify that Q shows that $\Phi ^{-1}$ is computable, and that Q is obtained uniformly in the given data. For a proof in a more general setting see [Reference Downey and Melnikov9, Theorem 4.2.48].

Proof. Suppose that u is a strong index for the set of strings $\{\alpha _1, \ldots , \alpha _r \} \subseteq T$ , and w is a strong index for the set of strings $\{\beta _1, \ldots , \beta _s \} \subseteq S$ . Let K be the subtree of T consisting of the prefixes, or extensions, of some $\alpha _i$ . Clearly one can uniformly obtain a computable bound H for this K as in Lemma 6.7. Let $n= \max _i |\beta _i|$ . Let $N= g(n)$ be the length computed from n through Lemma 6.7. Then $\Phi (\mathcal {K}^T_u) \subseteq \mathcal {K}^S_w$ if and only if for each $\alpha \in K$ of length N, there is an i such that $P_\Phi (\alpha ) \succeq \beta _i$ . By Lemma 6.7, this condition is decidable.

Proof. Given a string $\sigma \in T$ , if $\sigma = \emptyset $ then $\sigma \in \widetilde S$ . Assuming $\sigma \neq \emptyset $ , we can compute the least $t\in {\mathbb {N}}$ such that either $\sigma \prec f_t$ , or $\rho \not \in S$ for each $\rho \in T$ of length t such that $ \rho \succeq \sigma $ ; the latter condition can be decided by the hypotheses on T. Clearly $\sigma \in \widetilde S$ iff the former condition holds.

Proof. We use the notation of Definition 7.2. Let $n_i= | A_i|$ . By the hypotheses one can effectively identify the elements of $ A_i$ with $\{ 0, \ldots , n_i-1\}$ . Let

$$ \begin{align*} T = \{ \sigma \in {\mathbb{N}}^* \colon \forall i< |\sigma| [\sigma(i) < n_i \, \land \, [ i>0 \to \phi_i(\sigma(i)) = \sigma(i-1)]]\}. \end{align*} $$

It is clear from the hypotheses that T is c.l.c. via $k=0$ (Definition 2.4). To show that the binary group operation ${Mult}$ on $[T]$ is computable, for strings $\sigma _0, \sigma _1$ on T of the same length $k+1$ , let $P_2 (\sigma _0, \sigma _1)$ be the unique string $\tau \in T$ of length $k+1$ such that in $\mathcal {A}_k$ one has $\sigma _0(k) \sigma _1(k) = \tau (k)$ . The inversion operation ${Inv}$ on $[T]$ is computable by a similar argument.

Proof. “ $\Rightarrow $ ”: Let the computable set $D\subseteq {\mathbb {N}}$ be the domain of the computable presentation. For the computable Baire presentation, we take the c.l.c. tree

$$ \begin{align*}T_G = \{ r0^k \colon \, r \in D \, \land \, k \in {\mathbb{N}}\}, \end{align*} $$

and the operations canonically defined on $[T_G]$ via partial computable functions that only regard the first entry of the oracle strings.

“ $\Leftarrow $ ”: By Theorem 7.6 proved shortly below, G has a computable presentation via a meet groupoid. It now suffices to invoke Example 4.8.

Proof. (i) By Lemma 6.8 one can decide whether $\mathcal {K}_u\subseteq (\mathcal {K}_w)^{-1}$ . The equality $\mathcal {K}_u= (\mathcal {K}_w)^{-1}$ is equivalent to $\mathcal {K}_u\subseteq (\mathcal {K}_w)^{-1}\, \land \, \mathcal {K}_w\subseteq (\mathcal {K}_u)^{-1}$ . So one lets $I(u)$ be the least index v such that this equality holds.

(ii) Let $\widetilde T$ be the tree of initial segments of strings of the form $\sigma _0 \oplus \sigma _1$ , where $\sigma _0, \sigma _1 \in T$ have the same length. Then $\widetilde T$ is a c.l.c. tree, $[\widetilde T]$ is naturally homeomorphic to $[T] \times [T]$ , and ${Mult}$ can be seen as a computable function $[\widetilde T ] \to [T]$ . Now one applies Lemma 6.8.

Proof. “ $\Leftarrow $ ”: This follows from the corresponding implication in Theorem 5.1.

“ $\Rightarrow $ ”: We build a Haar computable copy $\mathcal {W}$ of the meet groupoid $\mathcal {W}(G)$ as required in Definition 4.6. By Lemma 7.5, one can decide whether $u \in E_T$ is the code number (Definition 2.5) of a subgroup. Furthermore, one can decide whether $B= \mathcal {K}_w $ is a left coset of a subgroup $U= \mathcal {K}_u$ : this holds iff $BU \subseteq B$ and $B^{-1} B \subseteq U$ , and the latter two conditions are decidable by Lemma 7.5. Similarly, one can decide whether B is a right coset of U.

It follows that the set $V= \{ w \in E_T \colon \, \mathcal {K}_w \text { is a coset}\}$ can be obtained via an existential quantification over a computable binary relation; in other words, V is r.e. A basic fact from computability theory states that from a description of V as an r.e. set one can uniformly obtain a computable set $\widetilde D \subseteq {\mathbb {N}}^+$ and a computable bijection $\theta \colon \widetilde D \to V$ . (Here $\widetilde D$ is the set of “stages” at which new elements are enumerated into V; one can assume that at each stage $n \in \widetilde D$ exactly one element $\theta (n)$ is enumerated.) Write $A_n = \mathcal {K}_{\theta (n)}$ for $n\in \widetilde D$ , and $A_0 = \emptyset $ .

The domain of $\mathcal {W}$ is $D = \widetilde D \cup \{0\}$ . By Lemma 2.6 the intersection operation on $\mathcal {W}$ is computable, i.e., there is a computable binary function c on ${\mathbb {N}}$ such that $A_{c(n,k)} =A_n \cap A_k$ . Given $n,k \in {\mathbb {N}}-\{0\}$ one can decide whether $A_n$ is a right coset of the same subgroup that $A_k$ is a left coset of. In that case, one can compute the number r such that $A_r= A_n\cdot A_k$ : one uses that $A_r$ is the unique coset C such that

1. (a) $A_nA_k \subseteq C$ , and

2. (b) C is a right coset of the same subgroup that $A_k$ is a right coset of.

As in the corresponding implication in the proof of Theorem 5.1, for subgroups $U,V$ , one can compute $|U \colon U \cap V|$ by finding in $\mathcal {W}$ further and further distinct left cosets of $U \cap V$ contained in U, until their union reaches U. The latter condition is decidable.

Notation 7.7. Given a computable Baire presentation G, by $\mathcal {W}_{\text {comp}} (G)$ we denote the computable copy of $\mathcal {W}(G)$ obtained in the proof above.

Proof. We provide the proof for the groups $\operatorname {\mathrm {SL}}_n({\mathbb {Q}}_p)$ , and then indicate the changes that are necessary for the groups $\operatorname {\mathrm {SL}}_n(\mathbb F_p((t)))$ . We begin by giving a computable Baire presentation of ${\mathbb {Q}}_p$ as a ring.

Let Q be the tree of strings $\sigma \in {\mathbb {N}}^*$ such that all the entries, except possibly the first, are among $\{0, \ldots , p-1\}$ , and $r0 \not \preceq \sigma $ for each $r>0$ . Clearly Q is c.l.c. via $k=1$ (Definition 2.4). We think of a string $r \hat {\ } \sigma \in Q$ as denoting the rational number $p^{-r} n_\sigma \in {\mathbb {Z}}[1/p]$ , where $n_\sigma $ is the natural number that has $\sigma $ as a p-ary expansion, written in reverse order:

$$ \begin{align*} n_\sigma = \sum_{i< |\sigma|} p^i \sigma(i). \end{align*} $$

The condition that $r0 \not \preceq \sigma $ for each $r>0$ expresses that p does not divide $n_\sigma $ unless $r=0$ . Let $\sigma , \tau $ be strings over $\{0, \ldots , p-1\}$ of the same length $\ell $ . By $\sigma + \tau $ we denote the string $\rho $ of the same length $\ell $ such that $n_\rho = n_\sigma + n_\tau \mod p^\ell $ . By $\sigma \cdot \tau $ we denote the string $\rho $ of length $2\ell $ such that $n_\rho = n_\sigma n_\tau \mod p^{2\ell }$ .

We now provide partial computable functions $P, P_1, P_2, P_3$ according to Remark 6.5 that are defined on computable subsets of the trees Q or $Q^2$ , in order to show that various operations on $[Q]$ are computable according to Definition 6.3. For a string of the form $r\sigma $ let

$$ \begin{align*} P(r\sigma )=( r-k)\tau, \end{align*} $$

where $k\in {\mathbb {N}} $ is maximal such that $k \le r$ and $0^k\preceq \sigma $ , and $\tau $ is the rest, i.e., $0^k \tau = \sigma $ . To show the computability of the function $q \to -q$ , let

$$ \begin{align*} P_1(r\sigma )= r\tau\ \text{where}\ |\tau |=|\sigma | \ \text{and}\ \sigma + \tau =0 \mod p^{|\sigma |}.\end{align*} $$

To show that the addition operation is computable, if $r, s \in {\mathbb {N}}$ and $\sigma , \tau $ are strings such that $r+ |\tau | = s + |\sigma |$ , let

$$ \begin{align*} P_2(r\sigma , s\tau ) = \begin {cases} P (s(0^{s-r} \sigma + \tau ) &\text {if } r \le s\\ P (r(\sigma + 0^{r-s} \tau )&\text {otherwise.} \end {cases}\end{align*} $$

To show that the multiplication operation is computable, let

$$ \begin{align*} P_3(r\sigma , s\tau ) = \begin {cases} P ((2s)(0^{s-r} \sigma \cdot \tau )) & \text {if } r \le s\\ P ((2r)(\sigma \cdot 0^{r-s} \tau )) & \text {otherwise. } \end {cases}\end{align*} $$

$P_2$ and $P_3$ are the correct functions because, say for $r\le s$ , in ${\mathbb {Z}}[1/p]$ one has $p^{-r} n_\sigma + p^{-s}n_\tau = p^{-s}(n_{0^{s-r}\sigma }+n_\tau )$ and $p^{-r} n_\sigma p^{-s}n_\tau = p^{-2s}n_{0^{s-r}\sigma }n_\tau $ .

We next provide a computable Baire presentation $([T], {Mult}, {Inv})$ of $\operatorname {\mathrm {SL}}_n({\mathbb {Q}}_p)$ as in Definition 7.1. Let T be the c.l.c. tree that is an $n^2$ -fold “power” of Q. More precisely, $T = \{ \sigma \colon \forall i < n^2 \, [ \sigma ^i \in Q]\}$ , where $\sigma ^i$ is the string of entries of $\sigma $ in positions of the form $\ell n^2 + i$ for some $\ell ,i \in {\mathbb {N}}$ . Clearly, $[T]$ is c.l.c. via $k= n^2$ , and T can be naturally identified with the matrix algebra $M_n({\mathbb {Q}}_p)$ . By the computability of the ring operations on ${\mathbb {Q}}_p$ as verified above, the matrix product is computable as a function $[T] \times [T]\to [T]$ , and the function $\det \colon [T] \to [Q]$ is computable.

Note that for any c.l.c. trees T and R, any computable path f of R, and any computable function $\Phi \colon [T]\to [R]$ , there is a computable subtree $S $ of T such that $[S]$ equals the pre-image $\Phi ^{-1}(f)$ . (In the language of computable analysis, the pre-image is effectively closed.) To see this, suppose that $\Phi $ is computable according to Definition 6.3 via a partial computable function L taking pairs in $T\times {\mathbb {N}}$ to ${\mathbb {N}}$ . Let S consist of the strings $\sigma \in T$ of length t such that t steps of the attempted computations $L(\sigma ,k) $ don’t yield a contradiction to the hypothesis that the output of the oracle Turing machine equals f. More formally,

$$ \begin{align*} S = \{ \sigma \in T \colon \forall k < |\sigma| [ L_{|\sigma|}(\sigma,k) \text{ is defined } \to L_{|\sigma|}(\sigma,k) = f(k)]\}. \end{align*} $$

In our setting, applying this to the function $\det \colon [T] \to [Q]$ and the path $f = 01000\ldots $ that denotes $1 \in {\mathbb {Q}}_p$ , we obtain a computable subtree S of T such that $[S]$ can be identified with $SL_n({\mathbb {Q}}_p)$ .

It is well-known that $SL_n({\mathbb {Z}}[1/p])$ is dense in $SL_n({\mathbb {Q}}_p)$ . This is a special case of strong approximation for algebraic groups (see [Reference Platonov and Rapinchuk45, Chapter 7]), but can also be seen in an elementary way using Gaussian elimination. The paths on S corresponding to matrices in $SL_n({\mathbb {Z}}[1/p])$ are precisely the ones that are $0$ from some point on. Clearly there is a computable listing $(f_i)$ of the set of such paths. So by Proposition 6.9 one can replace S by a c.l.c. tree $\widetilde S$ such that $[\widetilde S] = [S]$ .

To obtain a computable Baire presentation based on $\widetilde S$ , note that matrix multiplication on $[\widetilde S]$ is computable as the restriction of matrix multiplication on $[T]$ . To define the matrix inversion operation ${Inv}$ , we use the fact that the inverse of a matrix with determinant $1$ equals its adjugate matrix; the latter can be obtained by computing determinants on minors.

Next we treat $SL_n(\mathbb F_p((t)))$ . To give a Baire presentation of $\mathbb F_p((t))$ as a ring, we use the same tree Q as above, but now think of a string $r\sigma $ as representing the finite Laurent series $t^{-r} n_\sigma \in {\mathbb {Z}}[1/p]$ , where $n_\sigma = \sum _{i< |\sigma |} t^i \sigma (i)$ . For strings $\sigma , \tau $ over $\{0, \ldots , p-1\}$ and of the same length $\ell $ , by $\sigma + \tau $ we denote the string $\rho $ of length $\ell $ such that $n_\rho = n_\sigma + n_\tau \mod p^\ell $ . By $\sigma \cdot \tau $ we denote the string $\rho $ of length $2\ell $ such that $n_\rho \equiv n_\sigma n_\tau \mod t^{2\ell }$ . Now the ring operations are computable via the partial computable functions defined as for the case of ${\mathbb {Q}}_p$ , but with the new interpretation of the symbols “ $+$ ” and “ $\cdot $ ” as operations on strings. The remainder of the proof didn’t use any particulars about the computable Baire presentation of the ring ${\mathbb {Q}}_p$ , besides the fact that the path denoting $1$ is computable; this carries over to the present case. It is known that $SL_n(\mathbb F_p(t))$ , the group of matrices over finite Laurent series with determinant 1, forms a dense subgroup of $SL_n(\mathbb F_p((t)))$ , so from here on we can argue as before.

Proof. (i) To show that $\Phi $ is computable, we define a partial computable function $P_\Phi $ on T with values in S according to Remark 6.5. Given $\sigma \in T$ such that $[\sigma ]_T$ is compact, compute u such that $\mathcal {K}^T_u = [\sigma ]_T$ . Let $\{\beta _1, \ldots , \beta _s \} \subseteq S$ be the finite set with strong index $f(u)$ . Let $\eta =P_\Phi (\sigma )$ be the longest common initial segment of the strings $\beta _i$ . Note that with the standard ultrametric on $[S]$ , the set $[\eta ]_S$ is a closed ball centred at any element of $\mathcal {K}_{f(u)}^S$ , with the same diameter as $\mathcal {K}_{f(u)}^S$ . So if $\sigma \preceq \tau \in T$ then $[P_\Phi (\sigma )]_S \supseteq [P_\Phi (\tau )]_S$ . This shows that $P_\Phi $ is monotonic. Since $\Phi $ is continuous, one has

$$ \begin{align*} \Phi(f)= \bigcup_n \{ P_\Phi (f\! \upharpoonright_{n})\colon \, f\! \upharpoonright_{n} \in \mathrm{dom} (P_\Phi)\} \end{align*} $$

as required.

(ii) By Lemma 2.6 applied to S, there is a computable function g that is “inverse” to f in the sense such that for each v such that $\mathcal {K}_v^S$ is compact, one has $\Phi ^{-1}(\mathcal {K}_v^S)=\mathcal {K}_{g(v)}^T$ . Now one applies (i) to $\Phi ^{-1}$ and g.