Filling functions of arithmetic groups

Leuzinger, Enrico; Young, Robert M.

doi:10.4007/annals.2021.193.3.2

Cited by 3 publications

(2 citation statements)

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“…Can we have a coarse embedding from SL 2 (Z) × SL 2 (Z) into SL 3 (Z)? Note that Leuzinger and Young managed recently to give higher filling functions of some nonuniform lattices [LY21]. Question 6.5.…”

Section: Thereforementioning

confidence: 99%

Coarse embeddings of symmetric spaces and Euclidean buildings

Bensaid¹

2022

Preprint

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Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb and Eskin-Farb in the 90's. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. This is no longer the case for coarse embeddings as shown by horospherical embeddings. However, we show that in the absence of a Euclidean factor in the domain, the rank is monotonous under coarse embeddings. This answers a question by David Fisher and Kevin Whyte. We can also relax the condition on the domain by allowing it to contain a Euclidean factor of dimension 1, answering a question by Gromov. OUSSAMA BENSAID 4.4. Proof of Theorem 1.10 38 5. When the domain X has a one-dimensional Euclidean factor 49 5.1. Coarse embeddings of Euclidean spaces into lower rank 49 6. Further questions 54 References 55

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Section: Thereforementioning

confidence: 99%

Coarse embeddings of symmetric spaces and Euclidean buildings

Bensaid¹

2022

Preprint

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“…In the arithmetic case (i.e. if S consists of all infinite places) Leuzinger-Young [LY21] have established that the homological filling function changes from polynomial to exponential in the critical degree d. Thus while there is no qualitative change (i.e. affecting finiteness properties) visible in this case, there is a quantitative change (i.e.…”

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confidence: 99%

Higher finiteness properties of arithmetic approximate lattices: The Rank Theorem for number fields

Hartnick¹,

Witzel²

2022

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We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reductive approximate groups. For S-arithmetic approximate groups without infinite places we show that the finiteness length is finite and compute this finiteness length explicitly. In the simple case it is one less than the sum of the local ranks. This extends the Rank Theorem of Bux, Köhl and the second author from positive characteristic to characteristic zero. Our proof is based on a geometric version of their proof, but except for some input from reduction theory it is characteristic free. This indicates that the apparent differences between arithmetic groups in characteristic zero and positive characteristic concerning finiteness properties are entirely due to the presence of infinite places.

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