Math. Z. (2018) 288:287–298
Extremal elements in Coxeter groups and metric
commensurators of Kac-Moody groups
· Koen Struyve
Received: 14 July 2015 / Accepted: 24 April 2017 / Published online: 23 May 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We prove a characterization of irreducible, non-spherical and non-afﬁne Coxeter
groups, motivated by applications to metric commensurators of Kac-Moody groups and
twinnings at ﬁnite distance.
Let G be a Kac-Moody group over a ﬁnite ﬁeld. Then there is a natural action of G on
the product of two buildings
. The isometry group H of :=
equipped with a metric inherited from the metric on (the geometric realization of) and we
denote the image of G in H by Γ . In this paper we determine the metric commensurator
of Γ in H in the non-afﬁne case. We use the fact that Γ stabilizes a twinning δ
. The crucial observation is that an element in the metric commensurator of
Γ in H maps the twinning δ
onto a twinning at ﬁnite distance from δ
(see Sect. 4.2 for
the precise deﬁnition). The question about twinnings at ﬁnite distance yields naturally to
the deﬁnition of extremal elements in Coxeter systems. It turns out that the existence of an
extremal element in an irreducible Coxeter system implies that the latter is spherical or afﬁne.
Hence, as a byproduct of our investigations we obtain a new characterization of spherical
and afﬁne Coxeter systems. We ﬁrst present this characterization. Its proof does not require
any prerequisites from Kac-Moody theory and will be accomplished in Sect. 3. In Sect. 4 we
recall some basic facts about twinnings and investigate twinnings at ﬁnite distance. Finally,
Sect. 5 is about metric commensurators of Kac-Moody groups.
In the remainder of this introduction we describe our main results in more detail.
Mathematisches Institut, Arndtstraße 2, 35392 Gießen, Germany
Department of Mathematics, Ghent University, Krijgslaan 281, S22, 9000 Ghent, Belgium