Flows and joins of metric spaces

By Igor Mineyev. 2005

79 citations recorded by the source corpus.

We introduce the functor ◦ which assigns to every metric space X its symmetric join ◦X. As a set, ◦X is a union of intervals connecting ordered pairs of points in X. Topologically, ◦X is a natural quotient of the usual join of X with itself. We define an Isom(X)–invariant metric d∗ on ◦X. Classical concepts known for H n and negatively curved manifolds are defined in a precise way for any hyperbolic complex X, for example for a Cayley graph of a Gromov hyperbolic group. We define a double difference, a cross-ratio and horofunctions in the compactification ¯ X = X ⊔ ∂X. They are continuous, Isom(X)–invariant, and satisfy sharp identities. We characterize the translation length of a hyperbolic isometry g ∈ Isom(X). For any hyperbolic complex X, the symmetric join ◦ ¯ X of ¯ X and the (generalized) metric d∗ on it are defined. The geodesic flow space F(X) arises as a part of ◦ ¯ X. (F(X), d∗) is an analogue of (the total space of) the unit tangent bundle on a simply connected negatively curved manifold. This flow space is defined for any hyperbolic complex X and has sharp properties. We also give a construction of the asymmetric join X ◦ Y of two metric spaces. These concepts are canonical, i.e. functorial in X, and involve no “quasi”-language. Applications and relation to the Borel conjecture and others are discussed. AMS Classification numbers Primary: 20F65, 20F67, 37D40, 51F99, 57Q05 Secondary: 57M07, 57N16, 57Q91, 05C25

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0006862669163993d91ead9a438eac415d81037a

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Citations in the corpus

Symmetric Joins and Weighted Barycenters (2006, unknown): “A T ] 1 0 Ju n”
Dehn filling in relatively hyperbolic groups (2006, background): “Applications of Mineyev’s construction are myriad (see, for example, [22, 23, 24, 25, 26, 33]).”
Anosov representations: domains of discontinuity and applications (2011, methodology): “In order to extend the notion of Anosov representations to arbitrary finitely generated word hyperbolic groups Γ, we replace T 1Ñ by the flow space Γ̂, introduced by Gromov [37] and developed by Champetier [24], Mineyev [60] and others.”
Anosov representations and proper actions (2015, unknown): “+∞,ϕ−∞) : R\GΓ −→ ∂∞Γ 2∗ := {(η,η′) ∈ ∂ ∞Γ 2 | η6= η′} is a homeomorphism. The ﬂow space was constructed by Gromov [Gro87, Th.8.3.C], and more details were provided by Champetier [Cha94, §4]. Mineyev [Min05] introduced a diﬀerent construction of the ﬂow space of a hyperbolic graph with bounded valen”
The pressure metric for Anosov representations (2013, background): “Gromov [25] defined a geodesic flow U0Γ for a word hyperbolic group – that we shall call the Gromov geodesic flow – (see Champetier [17] and Mineyev [54] for details).”
The K-theoretic Farrell–Jones conjecture for hyperbolic groups (2007, methodology): “Namely, in order to show that hyperbolic groups fulfill Assumption 1.4 we build in [5] on Mineyevs [48] replacement of the geodesic flow and generalize the long and thin cells of Farrell-Jones for manifolds to certain covers of metric spaces.”
Anosov representations and dominated splittings (2016, methodology): “ppRdqsuch that: (transversality:) for every x˘yPB we have ˘pxq‘pyqRd, (equivariance:) for every P we have ˘p xqˆp q˘pxqand p xq ˆp qpxq. 6To get exact equality in pq we need the construction in [Mi], for which orbits of the ow are geodesics. If instead we use [Ch] or [Mat], the orbits of the ow are ”
Stable commutator length in word-hyperbolic groups (2006, result): “(3) Mineyev’s geodesic flow space ([29])”
Anosov subgroups: dynamical and geometric characterizations (2017, methodology): “This was Gromov’s original viewpoint, although not the one in [27].”
The Borel Conjecture for hyperbolic and CAT(0)-groups (2009, background): “The latter depends ultimately on work of Mineyev and Yu [43, 44].”
Eigenvalues and entropy of a Hitchin representation (2014, background): “In order to define the Anosov property for a convex representation, we need to recall the Gromov geodesic flow of Γ. Gromov [17] (see also Mineyev [28]) defines a proper cocompact action of Γ on ∂2Γ× R, which commutes with the action of R by translation on the final factor.”