Asymptotic cones are particular examples of ultralimits of metric spaces.
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All asymptotic cones of an hyperbolic space are real trees.
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In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group.
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Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.
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These arise for example as asymptotic cones of higher-rank symmetric spaces or as Bruhat Tits buildings of higher-rank groups over valued fields.
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Then any asymptotic cone of M is homeomorphic to a finite simplicial complex with top-dimensional simplices of dimension equal to the \ mathbb Q-rank of \ mathbf G.
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Such an isometric action gives rise ( via the choic of a principal ultrafilter ) to an action on the asymptotic cone of \ mathbb H ^ 2, which is a real tree.
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The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones ( or, more precisely, their bi-Lipschitz types ) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular.
9.
The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones ( or, more precisely, their bi-Lipschitz types ) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular.
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