Stephan Witzel - Arithmetic groups and their relatives

Abstract

Arithmetic groups are lattices on products of buildings and symmetric spaces. With appropriate assumptions the converse is true as well. I will discuss some of these assumptions and how their failure can lead to interesting groups.

Talk

We study a proper CAT(k) Raum space $X$ with extensible geodesics ($=$ complete?). $\Gamma \subseteq Isom(X)$ discrete, proper, cocompact.

$g\in \Gamma$ is rank one if it has an axis that does not bound a half plane. (axis?)

Theorem (Sisto-Osin): If $\Gamma$ contains a rank-one isometry it is acylindrically hyperbolic.

$\Gamma$ is higher rank if it contains no rank-one isometries.

Example: Let $S$ be a set of primes. ${\mathbb{Z}}_S=\mathbb{Z}[\frac{1}{p},p\in S]$, then we do some other stuff.

Ich habe ja gar keine Ahnung von Zahlentheorie. Wirklich mal höchste Zeit, dass ich mich in die Bücher einarbeite.

Question: if $/Gamma$ is higher rank, does $/Gamma$ have to be an $S$-arithmetic subgroup of a simple group? This is false but not a stupid question because the following theorems hold:

• Kleiner-Leeb, Stadler: X tends to be a product of symmetric spaces, Euclidean buildings and rank-one spaces
• Tits-Weiss: A locally compact Euclidean Building of dimension $\ge 3$ is Bruhat-Tits
• Margulis (?): If some condition isgiven the group is $S$-arithmetic.

Answer: No, $\Gamma$ could be of higher-rank and not $S$-arithmetic and:

• $X$ a produc of two trees (Burger-Mozes)
• $X$ a $2$-dimensional building that is not Bruhat-Tits but exotic.

Apart from that the following could conceivably happen:

• $X$ a product of $\ge 3$ factors
• $X$ a product of two factors, one of them higher rank
• $X$ Bruhat-Tits and $\Gamma$ a Galois-lattice?

Q2: Are they like arithmetic groups?

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