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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:
1.
How large is the set $$S$$?
2.
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.

##### MSC:
 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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##### References:
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