2.1. Complex hyperbolic 2-orbifolds

This paper will require some basic facts about torsion in complex hyperbolic lattices. An element of $\textrm {SU}(2,1)$ is elliptic if it has a fixed point in ${\mathbb{H}}^2_{\mathbb{C}}$ . The geometry of an elliptic element’s action on ${\mathbb{H}}^2_{\mathbb{C}}$ is reflected in its eigenvalues and how the eigenspaces behave under the hermitian form used to define $\textrm {SU}(2,1)$ :

1. (1) Suppose $g$ is elliptic with only two distinct eigenvalues.

   1. (a) If the $2$ -dimensional eigenspace is signature $(1,1)$ , then $g$ is a complex reflection acting by the identity on a totally geodesic ${\mathbb{H}}^1_{\mathbb{C}}$ in ${\mathbb{H}}^2_{\mathbb{C}}$ and by an element of ${\textrm {U}}(1)$ on its normal bundle.

   2. (b) If the $2$ -dimensional eigenspace is positive definite, then $g$ is a reflection through a point. In particular, it has a unique fixed point in ${\mathbb{H}}^2_{\mathbb{C}}$ and the action of $g$ on its tangent space, which is canonically identified with the rank $2$ eigenspace, is by a nontrivial scalar multiple of the identity.

2. (2) If $g$ is elliptic with three distinct eigenvalues, then it is regular elliptic. Here, $g$ has a unique fixed point in ${\mathbb{H}}^2_{\mathbb{C}}$ where the induced action on the tangent space to that point is by a nonscalar cyclic group.

Note that an element with only one eigenvalue is scalar, hence it acts trivially on ${\mathbb{H}}^2_{\mathbb{C}}$ . Of particular importance in this paper is the observation that an elliptic element of $\textrm {SU}(2,1)$ fails to have an isolated fixed point on ${\mathbb{H}}_{\mathbb{C}}^2$ if and only if it is a complex reflection.

Suppose $\Gamma \lt \textrm {SU}(2,1)$ is a lattice and $\Lambda \lt \Gamma$ is a torsion-free, normal subgroup of finite index. Then the finite group $F = \Gamma / \Lambda$ admits an isometric action on the manifold ${\mathbb{H}}_{\mathbb{C}}^2 / \Lambda$ . Whether or not $F$ has isolated fixed points is completely characterized by the torsion elements of $\Gamma$ .

Essential to this paper for the odd prime case of Theorem1.1 is that complex reflections and regular elliptic elements are distinguished by congruence quotients.

2.2. Arithmetic complex hyperbolic surfaces

Throughout this discussion, $k$ will be a totally real number field and $E/k$ a CM-extension (i.e., a totally imaginary quadratic extension of $k$ ). The rings of integers of these field will be denoted $R_k$ and $R_E$ , respectively. We will also fix an embedding of $E\hookrightarrow {\mathbb{C}}$ and refer to this as the identity embedding of $E$ ; note that complex conjugation restricts to the identity on $k$ .

We now recall the construction of arithmetic subgroups of $\textrm {SU}(2,1)$ , beginning with those of simplest type. Let $V$ be a $3$ -dimensional vector space over $E$ equipped with an $E$ -defined Hermitian form. That is, $H\in \mathrm{GL}(3,E)$ satisfies $H =\overline {H}^t$ . Furthermore, we also assume that $H$ has signature $(2,1)$ , and that for any nonidentity embedding $\nu : k \hookrightarrow {\mathbb{C}}$ and one (hence either) extension of $\nu$ to an embedding of $E$ , the matrix $H^\nu$ obtained by applying $\nu$ entry-wise has signature $(3,0)$ . We can and will assume that the entries of $H$ lie in $R_E$ . Define the algebraic group $\textrm {SU}(H)$ as

\begin{equation*} \textrm {SU}(H) = \{g\in \mathrm{SL}_3({\mathbb{C}}) \,:\, gH\overline {g}^t=H\}. \end{equation*}

The subgroup $\textrm {SU}(H,R_E) = \textrm {SU}(H) \cap SL(3,R_E)$ is an arithmetic lattice of $\textrm {SU}(H,{\mathbb{R}})$ that, after appropriate conjugation in $\mathrm{GL}(3,{\mathbb{C}})$ to take $H$ to the standard form, determines an arithmetic lattice in $\textrm {SU}(2,1)$ . The arithmetic lattices of simplest type are those determined by the totality of all commensurability classes of the groups described above. Finally, $\textrm {SU}(H, R_E)$ is cocompact if and only if $k \neq \mathbb{Q}$ .

The other arithmetic lattices are those of division algebra type. Continuing with the above notation, let $A$ be a division algebra over $E$ with center $E$ and degree three over $E$ , so then $A$ is a $9$ -dimensional $E$ -vector space. Suppose that $A$ also admits an involution $\tau$ such that the restriction $\tau |_E$ from the natural inclusion $E \hookrightarrow A$ is complex conjugation; such an involution is called an involution of the second kind. A Hermitian element of $A$ is an element $h\in A^*$ such that $\tau (h) = h$ . Now $A\otimes _E {\mathbb{C}} \cong M(3,{\mathbb{C}})$ and we insist that $h$ has signature $(2,1)$ and has signature $(3,0)$ after applying any embedding $\sigma : E\hookrightarrow {\mathbb{C}}$ extending a nonidentity embedding of $k$ .

Define the algebraic group $\textrm {SU}(h,A)$ as:

\begin{equation*} \textrm {SU}(h,A) =\! \left \{x \in A^1 \,:\, \tau (x)hx = h\right \}\!, \end{equation*}

where $A^1$ denotes the elements of $A$ with reduced norm $1$ ; note that the reduced norm becomes the determinant under any matrix embedding of $A$ . Let $\mathcal{O}\subset A$ be a $R_E$ -order of $A$ and define the arithmetic lattice $\textrm {SU}(h,\mathcal{O}) = \textrm {SU}(h,A) \cap \mathcal{O}$ . This then defines an arithmetic lattice in $\textrm {SU}(2,1)$ . The arithmetic lattices of division algebra type are those determined by the totality of all commensurability classes of the groups described above.

The next two subsections study torsion in arithmetic lattices of simplest type and division algebra type, respectively.

2.3. The groups $G(m,p,2)$

This section describes a special case of the results in [Reference McReynolds11] in some detail. Fix integers $m, p \ge 2$ such that $p$ divides $m$ . In what follows, $\mu _m$ will denote the group of all $m^{th}$ roots of unity. First, consider the abelian subgroup

\begin{equation*} A(m,p,2) =\!\left \{\begin{pmatrix} \xi _1 & 0 \\ 0 & \xi _2 \end{pmatrix}\ :\ \xi _1, \xi _2 \in \mu _m,\, (\xi _1 \xi _2)^{\frac {m}{p}} = 1 \right \} \end{equation*}

of ${\textrm {U}}(2)$ , which is normalized by the element $w \in {\textrm {U}}(2)$ of order two acting by coordinate permutations. Then $G(m,p,2) = A(m,p,2) \rtimes \langle w \rangle$ . Note that $G(m,m,2)$ is simply the dihedral group of order $2 m$ . The following records some standard facts about $G(m,p,2)$ .

Proof. The first statement is clear from the fact that diagonal matrices and $w$ all preserve the standard inner product on ${\mathbb{C}}^2$ . See [Reference Lehrer and Taylor8, § 2.2] for the second statement and [Reference Lehrer and Taylor8, Lemma 2.8] for the third and fourth. The fifth statement is simply that no nontrivial divisor of $p$ divides $2$ or $\frac {m}{p}$ . For the last statement, the element

(1) \begin{align} g_p = \left(\begin{array}{c@{\quad}c} \zeta _p & 0 \\ 0 & \zeta _p^{-1} \end{array}\right) \end{align}

is in $G(m,p,2)$ for any primitive $p^{th}$ root of unity $\zeta _p$ and has distinct eigenvalues if $p \ge 3$ .

In Proposition 2.6, complex reflection means the classical definition: an element of ${\textrm {U}}(2)$ with $1$ as an eigenvalue. However, this ends up having the same meaning once we let $G(m,p,2)$ act on ${\mathbb{H}}_{\mathbb{C}}^2$ . Specifically, take ${\textrm {U}}(2,1)$ with respect to the standard hermitian form $|z_1|^2 + |z_2|^2 - |z_3|^2$ on ${\mathbb{C}}^3$ , which by Proposition 2.6(1) gives an embedding

(2) \begin{align} \left(\begin{array}{c@{\quad}c} \xi _1 & 0 \\ 0 & \xi _2 \end{array}\right) \mapsto \left(\begin{array}{c@{\quad}c@{\quad}c} \xi _1 & 0 & 0 \\ 0 & \xi _2 & 0 \\ 0 & 0 & 1 \end{array}\right) \end{align}

of $A(m,p,2)$ in ${\textrm {U}}(2,1)$ . Note that $G(m,p,2)$ then embeds in ${\textrm {PU}}(2,1)$ under projection, so it acts faithfully on ${\mathbb{H}}^2_{\mathbb{C}}$ .

Convention: For the remainder of this paper, $G(m,p,2)$ is considered as embedded in ${\textrm {U}}(2,1)$ by Equation (2).

Fortunately, the two notions of complex reflection coincide.

The following last fact will be used in the sequel.

Corollary 2.8. Let $p$ be an odd prime, $m$ be divisible by $p$ , and suppose that $\Gamma \lt \textrm {SU}(2,1)$ contains

(3) \begin{equation} S(m, p, 2) = G(m, p, 2) \cap \textrm {SU}(2,1). \end{equation}

Then $\Gamma$ contains a regular elliptic element of order $p$ .

2.4. Torsion in lattices of division algebra type

Torsion in an arithmetic lattice of division algebra type is quite special. The first restriction on torsion in lattices of division algebra type is as follows.

The following generalizes an observation due to Prasad and Yeung [Reference Prasad and Yeung13, Lem. 9.2].

Proof. It suffices to consider $\Gamma$ a maximal lattice in $\textrm {SU}(2,1)$ . It is known that $\Gamma \cap A^1$ is a normal subgroup of $\Gamma$ with index a power of $3$ (see the Remark after Eq. (0) in [Reference Prasad and Yeung13]). Since $3 \mid \varphi (9)$ , it suffices to prove the lemma for $\tau \in A^1$ , which implies that $\tau ^n$ is central in $A$ . Moreover, the subalgebra $E(\tau )$ is a commutative subalgebra, hence it is a finite extension of $E$ . Note that $\tau \notin E$ , since otherwise it is central in $A$ . Therefore, $E(\tau ) / E$ is an extension of degree three.

Since $\tau ^n$ has reduced norm one, and the reduced norm on the center is simply $z \mapsto z^3$ , it follows that $\tau ^n$ is a $3^{rd}$ root of unity. In particular, $\tau$ is either an $n^{th}$ or $(3 n)^{th}$ root of unity, hence its minimal polynomial over $\mathbb{Q}$ has degree $\varphi (k n)$ for $k \in \{1,3\}$ . Since $\mathbb{Q}(\tau )$ is then Galois over $\mathbb{Q}$ , it follows that $\overline {E}(\tau )$ is Galois over the Galois closure $\overline {E}$ of $E$ over $\mathbb{Q}$ , with degree $1$ or $3$ .

First, suppose that $\overline {E}(\tau ) / \overline {E}$ is degree $3$ . The diagram

then consists only of Galois extensions. It is then a standard exercise in Galois theory to prove that $[\mathbb{Q}(\tau ) : \overline {E} \cap \mathbb{Q}(\tau )] = 3$ and that this degree divides $\varphi (k n)$ .

Now suppose that $E(\tau ) \subseteq \overline {E}$ . Let $G$ denote the Galois group of $E$ over $\mathbb{Q}$ , $G_E$ be the stabilizer of $E$ in $G$ , and $G_\tau$ be the stabilizer of the subfield $\mathbb{Q}(\tau )$ of $\overline {E}$ . Since $E(\tau ) / E$ is degree three, there is a diagram

of subgroups of $G$ . The image of $G_H$ in $G / G_\tau \cong \mathrm{Gal}(\mathbb{Q}(\tau ) / \mathbb{Q})$ is then $G_H / (G_H \cap G_\tau ) \cong \mathbb{Z} / 3 \mathbb{Z}$ . Thus $[\mathbb{Q}(\tau ) : \mathbb{Q}]$ is again divisible by three, as desired. Finally, note that if $n = p \gt 3$ is a prime not equal to $3$ , then $\varphi (3 p) = 2 (p-1)$ , hence $p \equiv 1 \pmod {3}$ for either value of $k$ .

In particular, one can construct many commensurability classes of lattices of division algebra type so that $ \textrm {SU}(h, \mathcal{O})$ is torsion-free. More germane to the purposes of this paper, Lemma 2.10 implies that it is impossible to prove Theorem3.1 for all primes using lattices of division algebra type. However, we can prove the following special case.

Sketch of the proof. Let $\zeta$ be a primitive $p^{th}$ root of unity. If $\varphi (p) = p-1$ is divisible by $3$ , then $L = \mathbb{Q}(\zeta )$ contains a subfield $E$ so that $L / E$ is Galois of degree $3$ . Then

\begin{equation*} k = E \cap \mathbb{Q}(\zeta + \zeta ^{-1}) \end{equation*}

is the maximal totally real subfield of $E$ , and $E / k$ is quadratic. Thus, $E / k$ is a CM-extension, since $E$ is Galois over $\mathbb{Q}$ and hence totally complex.

Let $A$ be a cyclic division algebra with center $E$ and involution of second kind containing $L$ as a maximal subfield. One can then construct an order $\mathcal{O}$ containing the ring of integers $R_L$ of $L$ . Choosing an appropriate hermitian element of $A$ produces a lattice $\Gamma \lt \textrm {SU}(2,1)$ so that $\zeta$ naturally lives in $\Gamma$ . Thus, $\Gamma$ contains an element of order $p$ . Fix any normal, torsion-free subgroup $\Lambda \lt \Gamma$ of finite index. By Lemma 2.9, all torsion elements of $\Lambda$ are regular elliptics, so the action of $\Gamma / \Lambda$ on ${\mathbb{H}}_{\mathbb{C}}^2 / \Lambda$ has only isolated fixed points by Lemma 2.2. There is an element of $\Gamma / \Lambda$ with order $p$ that has nontrivial fixed point set by construction, so this proves the proposition.

3.1. Odd primes

For the case of $p$ an odd prime, we will construct a lattice $\Gamma _p \lt \textrm {SU}(2,1)$ of simplest type containing a regular elliptic element of order $p$ . Consider a $p^{th}$ root of unity $\zeta _p$ , set $E = \mathbb{Q}(\zeta _p)$ , let $k$ be its maximal totally real subfield, and consider the identity embedding of $k$ sending a generator $\zeta _p + \zeta _p^{-1}$ to $2 \cos (2 \pi / p)$ .

Then $k$ has degree $d = \frac {p-1}{2}$ over $\mathbb{Q}$ , and it is a classical fact that the various real embeddings of $k$ embed $R_k$ as a lattice in ${\mathbb{R}}^d$ . In particular, there is an element $\beta \in R_k$ so that $\beta \lt 0$ under the identity embedding of $k$ and $\nu (\beta ) \gt 0$ for all other embeddings $\nu$ of $k$ . Thus the hermitian form $H_\beta$ given by $|z_1|^2 + |z_2|^2 + \beta |z_3|^2$ defines an arithmetic lattice of simplest type.

3.2. P = 2

To handle this case, we will make use of an example from [Reference Deraux, Parker and Paupert4] (see Theorem 1.2 therein with $p=3$ ). This example is discussed in [Reference Deraux, Parker and Paupert5] in more detail, where it is the group denoted by $\mathcal{S}(\overline {\sigma }_4,3)$ . Henceforth, we will denote this group by $\Gamma$ , and note that $\Gamma$ is cocompact. A presentation for $\Gamma$ is:

\begin{equation*} \Gamma =\! \left \langle R_1, R_2, R_3, J\,|\,R_1^3, (R_1J)^7, J^3, JR_1J^{-1}=R_2, JR_2J^{-1}=R_3, R_1R_2R_1R_2=R_2R_1R_2R_1 \right \rangle \!. \end{equation*}

See [Reference Deraux, Parker and Paupert5, Table 5.1] and note that, as stated after Table 5.1, the last two relations there can be omitted. Although not visible in the presentation given above, the element $R_1R_2$ has order $12$ (again, see [Reference Deraux, Parker and Paupert5, Tb. 5.1]), and so in particular there is an element in $\Gamma$ of order $2$ .

As shown in [Reference Deraux, Parker and Paupert4, Theorem 1.1], $\Gamma$ is arithmetic of simplest type with:

\begin{align*} k&=\mathbb{Q}(\sqrt {21}) & E&=\mathbb{Q}(\sqrt {-3},\sqrt {-7}) \end{align*}

The fact that $\Gamma$ is simplest type can be deduced from [Reference Deraux, Parker and Paupert4, § 6], and also follows from [Reference Stover14, Theorem 1.4] using the existence of a complex reflection in $\Gamma$ . As discussed in [Reference Deraux, Parker and Paupert4, § 6.1], $\Gamma$ can be conjugated to preserve a Hermitian form $H$ whose entries are algebraic integers in the field $E$ such that matrices representing $R_1$ and $J$ also have entries in $E$ . From the descriptions of $R_1$ and $J$ given in [Reference Deraux, Parker and Paupert5, p. 7] we see these matrices have determinant 1, and so from the presentation for $\Gamma$ we deduce that, since $R_2$ and $R_3$ are conjugates of $R_1$ , they also have determinant $1$ . Thus we can assume that $\Gamma \lt \textrm {SU}(H,R_E)$ .

Then [Reference Deraux, Parker and Paupert5, § 4.3.2 & Appx.] classifies the finite subgroups of $\Gamma$ , which are the complex reflection groups denoted $G_4$ and $G_5$ in the standard Shephard–Todd notation. Consulting [Reference Lehrer and Taylor8, Appx. D], all complex reflections in these groups have order $3$ . Thus, any element of order $2$ in $\Gamma$ is a reflection through a point. Applying the exact same argument as in Lemma 3.2 now produces manifolds $X_i$ with a $\mathbb{Z} / 2 \mathbb{Z}$ action having a nonempty set of isolated fixed points. This completes the case $p = 2$ , and thus completes the proof of Theorem3.1.

4.1. Fake projective planes

Cartwright and Steger’s completion of the classification of fake projective planes showed that they are all of division algebra type. They have Euler characteristic $3$ , which implies by Chern–Gauss–Bonnet that they are minimal-volume manifolds. Thus, any automorphism has fixed points, and they are isolated by Lemma 2.9 and Lemma 2.2. The automorphism groups of fake projective planes are also classified, and one sees actions of $\mathbb{Z}/p\mathbb{Z}$ for $p = 3,7$ [Reference Keum7].

4.2. The Cartwright–Steger surface

The Cartwright–Steger surface [Reference Cartwright and Steger2] is a minimal-volume compact complex hyperbolic $2$ -manifold, and it admits an action of $\mathbb{Z} / 3 \mathbb{Z}$ (e.g., see [Reference Stover15, § 5]). One can check directly that all torsion elements are reflections through points and regular elliptics, so the action has isolated fixed points. In contrast with fake projective planes, this example is arithmetic of simplest type. Moreover, one can show that some of the $3$ -torsion associated with isolated fixed points of the $\mathbb{Z} / 3 \mathbb{Z}$ action are reflections through points. Thus, this example is of a slightly different kind than those produced using the methods of this paper.

4.3. An example for $p = 5$

For the pair $E = \mathbb{Q}(\zeta _5)$ and $k = \mathbb{Q}(\sqrt {5})$ , one of the famous lattices $\Gamma _{\Sigma \mu }$ constructed by Deligne and Mostow [Reference Deligne and Mostow3] ends up being ${\textrm {PU}}(h, R_k)$ for the appropriate $h$ . If $\mathfrak{p}_5$ is the prime dividing $5$ and $\Gamma (\mathfrak{p}_5)$ is the congruence subgroup of $\Gamma _{\Sigma \mu }$ of level $\mathfrak{p}_5$ , then all torsion subgroups of $\Gamma (\mathfrak{p}_5)$ are isomorphic to $(\mathbb{Z} / 5 \mathbb{Z})^2$ generated by complex reflections. See [Reference Agol and Stover1, Prop. 3.12] and the surrounding discussion.

Note that each $(\mathbb{Z} / 5 \mathbb{Z})^2$ , while generated by complex reflections, contains regular elliptic elements of order $5$ . The congruence subgroup $\Gamma (\mathfrak{p}_5^2)$ of level $\mathfrak{p}_5^2$ is torsion-free, and thus admits an isometric action of $\mathbb{Z} / 5 \mathbb{Z}$ with nonempty set of isolated fixed points by Lemma 2.3 and the same argument as the proof of Lemma 3.3. One can show using known formulas for Euler characteristics of Deligne–Mostow orbifolds that ${\mathbb{H}}_{\mathbb{C}}^2 / \Gamma (\mathfrak{p}_5^2)$ has Euler characteristic $1875$ (cf. Question 5.1 below).