COMPUTABLY TOTALLY DISCONNECTED LOCALLY COMPACT GROUPS

Abstract We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentability of a t.d.l.c. group, and show their equivalence. In the process, we obtain an algorithmic Stone-type duality between t.d.l.c. groups and certain countable ordered groupoids given by the compact open cosets. Several natural groups, such as Aut ( T d ) $\mathrm {Aut}(T_d)$ upper A u t left parenthesis upper T Subscript d Baseline right parenthesis and SL n ( Q p ) $\mathrm {SL}_n(\mathbb Q_p)$ upper S upper L Subscript n Baseline left parenthesis double struck upper Q Subscript p Baseline right parenthesis , have computable presentations. We provide a criterion based on the duality when a computable presentation of a t.d.l.c. group is unique up to computable isomorphism. We show that many constructions leading from t.d.l.c. groups to new t.d.l.c. groups have algorithmic versions that stay within the class of computably presented t.d.l.c. groups; most prominently, quotients by computable closed normal subgroups. We study whether objects associated with computably t.d.l.c. groups are computable: the modular function, the scale function, and Cayley–Abels graphs in the compactly generated case.