Continuity of discrete homomorphisms of diffeomorphism groups

Hurtado, Sebastian

doi:10.2140/gt.2015.19.2117

Cited by 17 publications

(17 citation statements)

References 23 publications

(32 reference statements)

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“…Two years after that, Ghys asked whether embeddability of Diff 0 ( ) into Diff 0 ( ) implies that the dimension of must be at least as large as that of . This was answered positively by Hurtado in 2015 using his automatic continuity result [Hur15], improving on a very low-dimensional case I had proved earlier (but that case did not have automatic continuity!). Hurtado also gives a more precise picture of what a classification might look like.…”

Section: Structure Theorems For Group Actionsmentioning

confidence: 91%

The Structure of Homeomorphism and Diffeomorphism Groups

Mann¹

2021

Notices Amer. Math. Soc.

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My favorite opening to a math talk is from a 2014 lecture of Etienne Ghys. Part of a minicourse for young researchers in geometric group theory, he begins the lecture with "My second favorite group [dramatic pause...] is the group of all diffeomorphisms of a compact manifold." Beyond the obvious question what is your first favorite, then? (for this, one should see the rest of the lecture series), this line has another hook that I like even better. Ghys' statement is a bit like answering the question "what is your favorite food" with "my favorite food is dessert!" That's a great response, I couldn't agree more, but didn't you just cheat there by naming an infinite class of things in place of a single thing? For it has been known since the 1980s that varying the manifold and even varying what you mean by diffeomorphism (smooth, 1 , 2 ,...) produces an infinite

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Section: Structure Theorems For Group Actionsmentioning

confidence: 91%

The Structure of Homeomorphism and Diffeomorphism Groups

Mann¹

2021

Notices Amer. Math. Soc.

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“…where f n S denotes the word length of f n with respect to some finite generating set S for Γ. Though this definition appears somewhat artificial, it is useful: distortion has been shown to place strong constraints on the dynamics of a homeomorphism (see in particular [14]), and is used in [18] to relate the algebraic and topological structure of Diff 0 (M ).…”

Section: Distortion In Homeomorphism Groupsmentioning

confidence: 99%

Large-scale geometry of homeomorphism groups

Mann¹,

Rosendal²

2017

Ergod. Th. Dynam. Sys.

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Let M be a compact manifold. We show the identity component Homeo 0 (M ) of the group of self-homeomorphisms of M has a well-defined quasi-isometry type, and study its large scale geometry. Through examples, we relate this large scale geometry to both the topology of M and the dynamics of group actions on M . This gives a rich family of examples of non-locally compact groups to which one can apply the large-scale methods developed in previous work of the second author.

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“…For example, Kallman used this perspective to show that many "big" groups of homeomorphisms, such as the full homeomorphism group or diffeomorphism group of a manifold, admit a unique Polish (separable and completely metrizable) group topology [7]. Other instances of this algebraic-topological relationship can be seen in the main results of [4], [10], [6], and [11].…”

Section: Introductionmentioning

confidence: 99%

$\operatorname{PL}(M)$ admits no Polish group topology

Mann

2017

Fund. Math.

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We show that the group of piecewise linear homeomorphisms of any compact PL manifold does not admit a Polish group topology. This uses a) new results on the relationship between topologies on groups of homeomorphisms, their algebraic structure, and the topology of the underlying manifold, and b) new results on the structure of certain subgroups of PL(M ). The proof also shows that the group of piecewise projective homeomorphisms of S 1 has no Polish topology.

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Continuity of discrete homomorphisms of diffeomorphism groups

Cited by 17 publications

References 23 publications

The Structure of Homeomorphism and Diffeomorphism Groups

The Structure of Homeomorphism and Diffeomorphism Groups

Large-scale geometry of homeomorphism groups

$\operatorname{PL}(M)$ admits no Polish group topology

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