2 research outputs found
Computably totally disconnected locally compact groups
We study totally disconnected, locally compact (t.d.l.c.) groups from an
algorithmic perspective. We give various approaches to defining computable
presentations of t.d.l.c.\ groups, 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. We exploit the
flexibility given by these different approaches to show that several natural
groups, such as \Aut(T_d) and \SL_n(\QQ_p), have computable presentations.
We show that many construction 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. This leads to further examples, such as
\PGL_n(\QQ_p). 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 give a criterion when
computable presentations of t.d.l.c.~groups are unique up to computable
isomorphism, and apply it to \QQ_p as an additive group, and the semidirect
product \ZZ\ltimes \QQ_p