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On generators of arithmetic groups over function fields. (English) Zbl 1231.11047
Let \({\mathbb F}_q\) be the finite field with \(q\) elements, \(A={\mathbb F}_q[T]\) the polynomial ring, and \(F={\mathbb F}_q(T)\) the field of rational functions in an indeterminate \(T\). Let further \(D\) be a quaternion division algebra over \(k\) split at the place at infinity, \(\Lambda\) an \(A\)-order in \(D\), and \(\Gamma = \Lambda^{\ast}\) its multiplicative group.
Suppose that \(q\) is odd (it is known that quaternion algebras behave differently in characteristic 2), and let \({\mathfrak r} \in A\) be the discriminant of \(D\) (the product of the monic generators the primes in \(A\) where \(D\) ramifies). Then there exists \({\mathfrak a} \in A\) such that \(D\) is the \(F\)-algebra with basis vectors \(1,i,j,ij\) that satisfy the relations \[ i^2 = {\mathfrak a},\;j^2 = {\mathfrak r},\;ij = -ji. \] Now let \(\Lambda\) be the \(A\)-order \(\Lambda = A \oplus Ai\oplus Aj \oplus Aij\). Then \[ \Gamma = \{a+bi+cj+dij\mid a,b,c,d \in A;\;a^2-{\mathfrak a}b^2-{\mathfrak r}c^2+{\mathfrak a}{\mathfrak r}d^2 \in {\mathbb F}_q^{\ast}\}. \] In his main result Theorem 1.2, the author shows:
The finite set \[ S=\{\gamma \in \Gamma~|~\|\gamma\| \leq 4 \deg ({\mathfrak a}{\mathfrak r}) +6\} \] generates \(\Gamma\), where \(\|a+bi+cj+dij\|\) is the maximum of the degrees of \(a,b,c,d\).
The author further discusses related problems:
How large is the set \(S\)?
To what extent can the set \(S\) be reduced to a generating subset \(S'\) with a description still essentially independent of \(\Gamma\), i.e., with absolute constants \(\sigma,\delta\) such that \[ S' = \{\gamma \in \Gamma~|~\|\gamma\| \leq \sigma~\deg ({\mathfrak a}{\mathfrak r}) + \delta\}? \]
The proof of Theorem 1.2 uses the action of \(\Gamma\) on the attached Bruhat-Tits tree and a quantification of the discontinuity of this action.

11F06 Structure of modular groups and generalizations; arithmetic groups
11G18 Arithmetic aspects of modular and Shimura varieties
20E08 Groups acting on trees
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