Arithmetic hyperbolic 3-manifold

A quaternion algebra over a field ${\displaystyle F}$ is a four-dimensional central simple ${\displaystyle F}$-algebra. A quaternion algebra has a basis ${\displaystyle 1,i,j,ij}$ where ${\displaystyle i^{2},j^{2}\in F^{\times }}$ and ${\displaystyle ij=-ji}$.

A quaternion algebra is said to be split over ${\displaystyle F}$ if it is isomorphic as an ${\displaystyle F}$-algebra to the algebra of matrices ${\displaystyle M_{2}(F)}$; a quaternion algebra over an algebraically closed field is always split.

If ${\displaystyle \sigma }$ is an embedding of ${\displaystyle F}$ into a field ${\displaystyle E}$ we shall denote by ${\displaystyle A\otimes _{\sigma }E}$ the algebra obtained by extending scalars from ${\displaystyle F}$ to ${\displaystyle E}$ where we view ${\displaystyle F}$ as a subfield of ${\displaystyle E}$ via ${\displaystyle \sigma }$.

A subgroup of ${\displaystyle \mathrm {PGL} _{2}(\mathbb {C} )}$ is said to be derived from a quaternion algebra if it can be obtained through the following construction. Let ${\displaystyle F}$ be a number field which has exactly two embeddings into ${\displaystyle \mathbb {C} }$ whose image is not contained in ${\displaystyle \mathbb {R} }$ (one conjugate to the other). Let ${\displaystyle A}$ be a quaternion algebra over ${\displaystyle F}$ such that for any embedding ${\displaystyle \tau :F\to \mathbb {R} }$ the algebra ${\displaystyle A\otimes _{\tau }\mathbb {R} }$ is isomorphic to the Hamilton quaternions. Next we need an order ${\displaystyle {\mathcal {O}}}$ in ${\displaystyle A}$. Let ${\displaystyle {\mathcal {O}}^{1}}$ be the group of elements in ${\displaystyle {\mathcal {O}}}$ of reduced norm 1 and let ${\displaystyle \Gamma }$ be its image in ${\displaystyle M_{2}(\mathbb {C} )}$ via ${\displaystyle \phi }$. We then consider the Kleinian group obtained as the image in ${\displaystyle \mathrm {PGL} _{2}(\mathbb {C} )}$ of ${\displaystyle \phi ({\mathcal {O}}^{1})}$.

The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on ${\displaystyle \mathrm {PGL} _{2}(\mathbb {C} )}$. Moreover, the construction above yields a cocompact subgroup if and only if the algebra ${\displaystyle A}$ is not split over ${\displaystyle F}$. The discreteness is a rather immediate consequence of the fact that ${\displaystyle A}$ is only split at its complex embeddings. The finiteness of covolume is harder to prove.[1]

An arithmetic Kleinian group is any subgroup of ${\displaystyle \mathrm {PGL} _{2}(\mathbb {C} )}$ which is commensurable to a group derived from a quaternion algebra. It follows immediately from this definition that arithmetic Kleinian groups are discrete and of finite covolume (this means that they are lattices in ${\displaystyle \mathrm {PGL} _{2}(\mathbb {C} )}$).

Examples are provided by taking ${\displaystyle F}$ to be an imaginary quadratic field, ${\displaystyle A=M_{2}(F)}$ and ${\displaystyle {\mathcal {O}}=M_{2}(O_{F})}$ where ${\displaystyle O_{F}}$ is the ring of integers of ${\displaystyle F}$ (for example ${\displaystyle F=\mathbb {Q} (i)}$ and ${\displaystyle O_{F}=\mathbb {Z} [i]}$). The groups thus obtained are the Bianchi groups. They are not cocompact, and any arithmetic Kleinian group which is not commensurable to a conjugate of a Bianchi group is cocompact.

If ${\displaystyle A}$ is any quaternion algebra over an imaginary quadratic number field ${\displaystyle F}$ which is not isomorphic to a matrix algebra then the unit groups of orders in ${\displaystyle A}$ are cocompact.

For the volume an arithmetic three manifold ${\displaystyle M=\Gamma _{\mathcal {O}}\backslash \mathbb {H} ^{3}}$ derived from a maximal order in a quaternion algebra ${\displaystyle A}$ over a number field ${\displaystyle f}$ we have the expression:[2]

${\displaystyle \mathrm {vol} (M)={\frac {2|D_{F}|^{\frac {3}{2}}\cdot \zeta _{F}(2)}{2^{2r+1}\cdot \pi ^{2r}}}\cdot \prod _{{\mathfrak {p}}|D_{A}}(N({\mathfrak {p}})-1).}$

where ${\displaystyle D_{A},D_{F}}$ are the discriminants of ${\displaystyle A,F}$ respectively, ${\displaystyle \zeta _{F}}$ is the Dedekind zeta function of ${\displaystyle F}$ and ${\displaystyle r=[F:\mathbb {Q} ]}$.

A consequence of the volume formula in the previous paragraph is that

Given ${\displaystyle v>0}$ there are at most finitely many arithmetic hyperbolic 3–manifolds with volume less than ${\displaystyle v}$.

This is in contrast with the fact that hyperbolic Dehn surgery can be used to produce infinitely many non-isometric hyperbolic 3-manifolds with bounded volume. In particular, a corollary is that given a cusped hyperbolic manifold, at most finitely many Dehn surgeries on it can yield an arithmetic hyperbolic manifold.

The Weeks manifold is the hyperbolic three-manifold of smallest volume[3] and the Meyerhoff manifold is the one of next smallest volume.

The complement in the three—sphere of the figure-eight knot is an arithmetic hyperbolic three—manifold[4] and attains the smallest volume among all cusped hyperbolic three-manifolds.[5]

The Ramanujan conjecture for automorphic forms on ${\displaystyle \mathrm {GL} (2)}$ over a number field would imply that for any congruence cover of an arithmetic three-manifold (derived from a quaternion algebra) the spectrum of the Laplace operator is contained in ${\displaystyle [1,+\infty )}$.