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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The $\Sigma$-invariants of $S$-arithmetic subgroups of Borel groups
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by Eduard Schesler HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 4173-4237 Request permission


Given a Chevalley group $\mathcal {G}$ of classical type and a Borel subgroup $\mathcal {B} \subseteq \mathcal {G}$, we compute the $\Sigma$-invariants of the $S$-arithmetic groups $\mathcal {B}(\mathbb {Z}[1/N])$, where $N$ is a product of large enough primes. To this end, we let $\mathcal {B}(\mathbb {Z}[1/N])$ act on a Euclidean building $X$ that is given by the product of Bruhat–Tits buildings $X_p$ associated to $\mathcal {G}$, where $p$ is a prime dividing $N$. In the course of the proof we introduce necessary and sufficient conditions for convex functions on $CAT(0)$-spaces to be continuous. We apply these conditions to associate to each simplex at infinity $\tau \subset \partial _\infty X$ its so-called parabolic building $X^{\tau }$ and to study it from a geometric point of view. Moreover, we introduce new techniques in combinatorial Morse theory, which enable us to take advantage of the concept of essential $n$-connectivity rather than actual $n$-connectivity. Most of our building theoretic results are proven in the general framework of spherical and Euclidean buildings. For example, we prove that the complex opposite each chamber in a spherical building $\Delta$ contains an apartment, provided $\Delta$ is thick enough and $Aut(\Delta )$ acts chamber transitively on $\Delta$.
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Additional Information
  • Eduard Schesler
  • Affiliation: Fakultät für Mathematik und Informatik, FernUniversität in Hagen, 58084 Hagen, Germany
  • MR Author ID: 1496107
  • Email:
  • Received by editor(s): April 24, 2022
  • Received by editor(s) in revised form: November 7, 2022, and November 9, 2022
  • Published electronically: January 27, 2023
  • Additional Notes: The author was supported by the DFG grant WI 4079/4 within the SPP 2026 Geometry at infinity.
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 4173-4237
  • MSC (2000): Primary 20F65, 20E42, 51E24, 20F16
  • DOI:
  • MathSciNet review: 4586809