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 $\mathrm {Aut}(T_d)$ and $\mathrm {SL}_n(\mathbb Q_p)$, 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.

We investigate what it means for a (Hausdorff, second-countable) topological group to be computable. We compare several potential definitions based on classical notions in the literature. We relate these notions with the well-established definitions of effective presentability for discrete and profinite groups, and compare our results with similar results in computable topology.

In this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic $2$, the Tits index of simple algebraic subgroups of $\operatorname{Aut}(T)$ is not always preserved under Chabauty limits.

We establish various properties of the definition of cohomology of topological groups given by Grothendieck, Artin and Verdier in SGA4, including a Hochschild–Serre spectral sequence and a continuity theorem for compact groups. We use these properties to compute the cohomology of the Weil group of a totally imaginary field, and of the Weil-étale topology of a number ring recently introduced by Lichtenbaum (both with integer coefficients).