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Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows
Anosov flows have a long and rich history, firstly motivated by the
study of geodesic flows in negative curvature surface by Anosov and Sinai.
Not every closed manifold admits an Anosov flow for well-known reasons:
the fundamental group of a 3-manifold M admitting an Anosov flow must
have exponential growth, and M must be universally covered by R3. Nevertheless,
there are sufficient mechanisms for constructing distinct Anosov
flows on admissible 3-manifolds, such as Dehn-Goodman-Fried surgery or
playing with hyperbolic building blocks. A central problem in the field
has been to determine the number of Anosov flows that can be supported
by a single manifold. The question of whether there exists an infinite set
of pairwise non-equivalent Anosov flows on a 3-manifold remains open to
this day. However, there are several papers proving the existence of a
manifold Mn that admits n pairwise inequivalent Anosov flows for any
natural number n. In all known examples, the manifolds Mn are composed
of several geometric pieces. In the present paper, we prove the
existence of a countable number of graph manifolds Mk,n, k ∈ N with
a single geometric piece, each of which admits n pairwise non-equivalent
transitive Anosov flows. All previously known constructions of different
flows on the same graph manifold were based on gluing geodesic flows.
The nature of the flows constructed in this paper is completely different;
they are constructed from a single hyperbolic plug, which is a suspension
over a Morse-Smale diffeomorphism on a surface.