DTC ultrafilters on groups
- PDF / 278,350 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 20 Downloads / 195 Views
DTC ultrafilters on groups Jan Pachl1 · Juris Steprans ¯ 1 Received: 25 May 2020 / Accepted: 9 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We say that an ultrafilter on an infinite group G is DTC if it determines the topological centre of the semigroup βG. If G has a subgroup of finite index in which conjugacy classes are all finite and uniformly bounded in size, then G does not admit a DTC ultrafilter. On the other hand, if G has no subgroup of finite index in which all conjugacy classes are finite, then G does admit a DTC ultrafilter. It follows that an infinite finitely generated group admits a DTC ultrafilter if and only if it has no abelian subgroup of finite index. Keywords Ultrafilter · Topological centre · Virtually abelian group · FC group
1 Introduction ˇ When G is an infinite discrete group, its binary group operation extends to the Cech– Stone compactification βG in two natural ways. They are defined in Sect. 2 and, as in [3, Ch.6], denoted by and ♦. Say that v∈βG is a DTC ultrafilter for βG if uv = u♦v for every u∈βG\G. DTC stands for determining the (left) topological centre. This is a special case of a small set determining the topological centre of βG [3, Def.12.4]. Dales et al. [3] investigate the DTC notion in the context of their study of Banach algebras on semigroups and their second duals. They prove that the free group F2 admits a DTC ultrafilter [3, 12.22], and that no abelian group does. Here we address the problem of characterizing those groups that admit DTC ultrafilters, the class of groups we call DTC(1). It is not obvious how good such a characterization can be, even for countable groups, as it is not clear from the definition whether the class of
Communicated by Anthony To-Ming Lau. The research of Juris Stepr¯ans is supported by NSERC.
B 1
Jan Pachl [email protected] Department of Mathematics and Statistics, York University, Toronto, ON, Canada
123
J. Pachl, J. Stepr¯ans
countable DTC(1) groups, suitably encoded, is even within the projective hierarchy. However, it follows from our results that the class of finitely generated DTC(1) groups is a Borel set. In fact, we prove that an infinite finitely generated group is virtually abelian if and only if it does not belong to DTC(1). This provides a partial answer to question (13) in [3, Ch.13]. The algebraic property used above to define a DTC ultrafilter is equivalent to a topological one: v∈βG is a DTC ultrafilter for βG if and only if for every u∈βG\G the mapping w → uw from G ∪ {v} to βG is discontinuous at v. In other words, if and only if for every u∈βG the property • the mapping w → uw from G ∪ {v} to βG is continuous at v implies that u∈G and therefore u is in the topological centre of βG. The equivalence of the algebraic and the topological property is noted in the comment after Definition 12.4 in [3]. In this paper we do not deal with another notion of sets determining the topological centre, in which for every u∈βG\G the mapping w → uw from the whole βG to βG is requir
Data Loading...
