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Small diameters and generators for arithmetic lattices in $$\textrm{SL}_2(\mathbb {R})$$ and certain Ramanujan graphs
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AbstractWe show that arithmetic lattices in $$\textrm{SL}_{2}(\mathbb {R})$$
SL
2
(
R
)
, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $$\mathbb {Q}$$
Q
, admit a ‘small’ covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.
Title: Small diameters and generators for arithmetic lattices in $$\textrm{SL}_2(\mathbb {R})$$ and certain Ramanujan graphs
Description:
AbstractWe show that arithmetic lattices in $$\textrm{SL}_{2}(\mathbb {R})$$
SL
2
(
R
)
, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $$\mathbb {Q}$$
Q
, admit a ‘small’ covering set.
In particular, we give bounds on the diameter if the quotient space is co-compact.
Consequently, we show that these lattices admit small generators.
Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.
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