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Graphically discrete groups and rigidity
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DOI：10.1112/topo.70059.png)
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Abstract
En 中文
We introduce the notion of graphical discreteness to group theory. A finitely generated group is graphically discrete if whenever it acts geometrically on a locally finite graph, the automorphism group of the graph is compact-by-discrete. Notable examples include finitely generated nilpotent groups, most lattices in semisimple Lie groups, and irreducible nongeometric 3-manifold groups. We show graphs of groups with graphically discrete vertex groups frequently have strong rigidity properties. We prove free products of one-ended virtually torsion-free graphically discrete groups are action rigid within the class of virtually torsion-free groups. We also prove quasi-isometric rigidity for many hyperbolic graphs of groups whose vertex groups are closed hyperbolic manifold groups and whose edge groups are nonelementary quasi-convex subgroups. This includes the case of two hyperbolic 3-manifold groups amalgamated along a quasi-convex malnormal non-abelian free subgroup. We provide several additional examples of graphically discrete groups and illustrate this property is not a commensurability invariant.
Keywords:
QUASI-ISOMETRIES
COMBINATION THEOREM
MEASURE EQUIVALENCE
CONVERGENCE GROUPS
POLYNOMIAL-GROWTH
HYPERBOLIC GROUPS
COMMENSURABILITY
GEOMETRY
GRAPHS
AUTOMORPHISMS
Journal
J
IF:
1.1
Papers:
34
Citations:
0