Tree approximation in quasi-trees

Kerr, Alice

doi:10.48550/arxiv.2012.10741

Cited by 5 publications

(5 citation statements)

References 13 publications

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“…Delzant and Steenbock remark in their paper that if the space being acted on is a quasi-tree, then the logarithm terms in Theorem 1.5 should disappear (see [DS20,Remark 1.18]). This is due to the fact that tree approximation, a key ingredient in their proof, is uniform in the case of quasi-trees [Ker20]. We are able to check that this improvement does indeed remove the logarithm terms.…”

Section: Resultsmentioning

confidence: 90%

Product set growth in mapping class groups

Kerr¹

2021

Preprint

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The main theorem in this paper is that for any group G that acts acylindrically on a quasi-tree, there exists α > 0 such that for any finite U ⊂ G that has large enough displacement in the quasi-tree, and is not contained in a virtually cyclic subgroup, we have. By a result of Balasubramanya [Bal17], the main theorem applies to every acylindrically hyperbolic group. In particular, we can show that if G is a right-angled Artin group, then there exist α, β > 0 such that for any finite symmetric U ⊂ G that is not contained in Z, and not contained non-trivially in a subgroup of the form H × Z, we have |U n | (α|U |) βn .

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Section: Resultsmentioning

confidence: 90%

Product set growth in mapping class groups

Kerr¹

2021

Preprint

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“…The notion of a distributive waterfall map should generalise naturally to the case of maps between R-trees. Further, applying the result of Kerr [14] which states that quasi-trees are rough isometric to R-trees, there should be a well-defined notion of a distributive waterfall map between quasi-trees (i.e. one that is conjugate to a distributive waterfall map between two associated R-trees).…”

Section: W P Are the Vertices Connected By A Single Edge To V Such Th...mentioning

confidence: 99%

Embeddings of trees, cantor sets and solvable Baumslag–Solitar groups

Nairne

2022

Geom Dedicata

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We characterise when there exists a quasiisometric embedding between two solvable Baumslag–Solitar groups. This extends the work of Farb and Mosher on quasiisometries between the same groups. More generally, we characterise when there can exist a quasiisometric embedding between two treebolic spaces. This allows us to determine when two treebolic spaces are quasiisometric, confirming a conjecture of Woess. The question of whether there exists a quasiisometric embedding between two treebolic spaces turns out to be equivalent to the question of whether there exists a bilipschitz embedding between two symbolic Cantor sets, which in turn is equivalent to the question of whether there exists a rough isometric embedding between two regular rooted trees. Hence we answer all three of these questions simultaneously. It turns out that the existence of such embeddings is completely determined by the boundedness of an intriguing family of integer sequences.

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“…Remark 3.1. In [8], Kerr shows that any quasi-tree X is (1, C)-quasi-isometric to an R-tree T X . The definition of the metric d * on T X ([8, Proposition 4.2]) is very similar to our construction of d a .…”

Section: A Family Of Conditionally Negative Definite Kernelsmentioning

confidence: 99%

“…Observe that quasi-trees are not locally finite graphs in general. For a more detailed discussion, see [10] and [8]. We say that a metric space (X, d) is a finite product of quasi-trees if it decomposes as X = X 1 × • • • × X N , where each (X i , d i ) is a quasi-tree, and d is the ℓ 1 -metric:…”

Section: Introductionmentioning

confidence: 99%