Right-angled Coxeter groups with totally disconnected Morse boundaries. (English) Zbl 1522.20174
The Davis complex is a simplicial complex and metric space associated with any Coxeter group, generalizing the construction of Weyl chambers for finite Weyl groups or the standard apartment of the Bruhat-Tits building of an affine Weyl group. From the view of metric geometry, the Davis complexes provide an useful family of \(\mathrm{CAT}(0)\)-spaces.
The Morse boundary is an invariant of \(\mathrm{CAT}(0)\) metric spaces under quasi-isometry. As a subspace of the visual boundary, the Morse boundary has a clear geometric interpretation. Moreover, at least the visual boundary of the Davis complex has proved to be very useful for the structure theory of Coxeter groups (cf.the works of Timothée Marquis).
In this paper, the topological properties of the Morse boundary are studied for those Coxeter groups which are right angled, i.e.whose Coxeter graph only contains \(\infty\)-labelled edges. They show that in many cases, the topology of the Morse boundary of a right-angled Coxeter group can be described in terms of the topology of the Morse boundary of certain proper parabolic subgroups. Iterating this procedure, they produce a large number of examples of right-angled Coxeter groups whose Morse boundary is totally disconnected.
The Morse boundary is an invariant of \(\mathrm{CAT}(0)\) metric spaces under quasi-isometry. As a subspace of the visual boundary, the Morse boundary has a clear geometric interpretation. Moreover, at least the visual boundary of the Davis complex has proved to be very useful for the structure theory of Coxeter groups (cf.the works of Timothée Marquis).
In this paper, the topological properties of the Morse boundary are studied for those Coxeter groups which are right angled, i.e.whose Coxeter graph only contains \(\infty\)-labelled edges. They show that in many cases, the topology of the Morse boundary of a right-angled Coxeter group can be described in terms of the topology of the Morse boundary of certain proper parabolic subgroups. Iterating this procedure, they produce a large number of examples of right-angled Coxeter groups whose Morse boundary is totally disconnected.
Reviewer: Felix Schremmer (Hong Kong)
MSC:
| 20F65 | Geometric group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 57M07 | Topological methods in group theory |
| 57M15 | Relations of low-dimensional topology with graph theory |
Keywords:
RACGs; Morse boundary; \(\mathrm{CAT}(0)\) space with block decomposition; amalgamation; contracting boundaryReferences:
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