Homeomorphic subsurfaces and the omnipresent arcs

Fanoni, Federica; Ghaswala, Tyrone; McLeay, Alan

doi:10.48550/arxiv.2003.04750

Cited by 3 publications

(5 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let F ⊂ Ends(S) be the subset of ends with a finite Map(S)-orbit. Following [14] we say that a surface is stable if for any end e ∈ Ends(S) there exists a sequence of nested subsurfaces U 1 ⊃ U 2 ⊃ . .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…If F is empty then we the result follows from Theorem 4.1. As in the previous subsection, since S is stable we have that |F| = n is finite as otherwise there would be an accumulation point without a stable neighborhood [14]. Therefore we have a partition of Ends(S)…”

Section: 2mentioning

confidence: 92%

“…We say that an end e ∈ Ends(S) is of higher rank than e ∈ Ends(S) if each stable neighborhood of e contains an element of the Map(S)-orbit of e . Finally, we define F ⊂ Ends(S) to be the set of ends whose Map(S)-orbit is finite [14].…”

Section: 2mentioning

confidence: 99%

See 2 more Smart Citations

Big mapping class groups and the co-Hopfian property

Aramayona¹,

Leininger²,

McLeay³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the first examples of injective endomorphisms of mapping class groups (of surfaces with empty boundary) that fail to be surjective.We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding.Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 92%

See 1 more Smart Citation

Big mapping class groups and the co-Hopfian property

Aramayona¹,

Leininger²,

McLeay³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Hence the statement and some arguments are cleaner with this assumption. The class of stable surfaces, first used in [8], is a large and natural class of surfaces to work with and it includes all easily constructed infinite type surfaces.…”

Section: Theorem 15 (Topological Characterization Of An Essential Shi...mentioning

confidence: 99%

Asymptotic Dimension of Big Mapping Class Groups

Grant,

Rafi,

Verberne

2021

Preprint

View full text Add to dashboard Cite

Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasiisometry type. We show that the big mapping class group of a stable surface of infinite type with a coarsely bounded generating set that contains an essential shift has infinite asymptotic dimension. This is in contrast with the mapping class groups of surfaces of finite type where the asymptotic dimension is always finite. We also give a topological characterization of essential shifts.

show abstract

“…We note that in the days this survey was being finalized, Fanoni-Ghaswala-M c Leay [42] constructed new examples of hyperbolic infinite-diameter graphs that admit actions of big mapping class groups with unbounded orbits. We direct the reader to their article for details.…”

Section: Geometric Aspectsmentioning

confidence: 99%