Symmetries of exotic negatively curved manifolds

TL;DR

This paper investigates the symmetry properties of manifolds that are topologically but not smoothly equivalent to hyperbolic manifolds, providing examples with both maximal and minimal symmetry groups.

Contribution

It constructs examples of exotic negatively curved manifolds exhibiting either maximal or minimal symmetry, using advanced techniques like smoothing theory and hyperbolic manifold construction.

Findings

Existence of manifolds with arbitrarily large symmetry groups.

Existence of manifolds with minimal symmetry, no small subgroup acts by diffeomorphisms.

Demonstrates the range of symmetry behaviors in exotic negatively curved manifolds.

Abstract

Let $N$ be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold $M$. In this paper, we study the extent to which $N$ admits as much symmetry as $M$. Our main results are examples of $N$ that exhibit two extremes of behavior. On the one hand, we find $N$ with maximal symmetry, i.e. Isom($M$) acts on $N$ by isometries with respect to some negatively curved metric on $N$. For these examples, Isom($M$) can be made arbitrarily large. On the other hand, we find $N$ with little symmetry, i.e. no subgroup of Isom($M$) of "small" index acts by diffeomorphisms of $N$. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group.

Equations

$$s(N)=\underset{\rho }{\sup }\frac{|\mathrm{Isom}(N,\rho )|}{|\mathrm{Isom}(M)|},$$

$$\frac{1}{|\mathrm{Isom}(M)|}\le s(N)\le 1$$

$$s(\Sigma ):=\underset{\rho }{\sup }\frac{\dim \mathrm{Isom}(\Sigma ,\rho )}{\dim \mathrm{Isom}(S^n)},$$

$$\mathrm{Diff}(N)$$

$$s(N)=\underset{F}{\max }\frac{|F|}{|\mathrm{Out}(\pi )|},$$

$$\mathrm{InjRad}(\overline{M})\ge \mathrm{InjRad}(M)/|F|>{\tau }_n.$$

$$1\to K\to \Delta \to F_r\to 1$$

$$N_{\Gamma }(\pi )/\pi \simeq N_{\mathrm{Isom}(H^n)}(\pi )/\pi \simeq \mathrm{Isom}(H^n/\pi ).$$

$${\mathrm{SO}}_{n+1}(\mathbb{C})/{\mathrm{SO}}_0(n,1)\to B{\mathrm{SO}}_0(n,1)\to B{\mathrm{SO}}_{n+1}(\mathbb{C})$$

$${\overline{\alpha }}_{\mathfrak{p},\mathfrak{q}}(g)=({\overline{\alpha }}_{\mathfrak{p}}(g),{\overline{\alpha }}_{\mathfrak{q}}(g))$$

$$\operatorname{SL}_{m}(\mathfrak{k})=\ker\big{[}\operatorname{SL}_{m}(\mathcal{O}_{S})\rightarrow\operatorname{SL}_{m}(\mathcal{O}_{S}/\mathfrak{k}\mathcal{O}_{S})\big{]}$$

$$R\ge \frac{1}{4}d(y,\eta y).$$

$$d(y,\eta y)\ge (1/c_1)\cdot w(\eta )-1.$$

$$w(\eta )\ge c_2\log k+c_3$$

$$|X{|}_s=\underset{i,j}{\max }s(x_{ij})\text{ and }|X{|}_S=\sum _{s\in S}|X{|}_s.$$

$$\ell \cdot k^{1/\ell }-\ell \le |\eta {|}_S\le m^{w(\eta )-1}\cdot M^{w(\eta )},$$

$$\prod _{s\in S}s(a/b)=1\text{ and }\prod _{s\in S}s(x_1)=N(x_1).$$

$$|\eta {|}_S\ge |{\eta }_{ij}{|}_S\ge \ell \cdot k^{1/\ell }-\ell .$$

$$S^1\times D^{n-1}\coprod M\setminus \iota (S^1\times \text{ int}(D^{n-1}))$$

$$\hat{\iota }:M_{+}\to M\stackrel{{\iota }^{\prime }}{\to }\underset{\ell }{\bigvee }{\Sigma }^{n-1}(S_{+}^1)\to \underset{\ell }{\bigvee }{S}^{n-1},$$

$$\hat{\iota}^{*}:\big{[}\bigvee_{\ell}S^{n-1},\>\operatorname{Top}/O\big{]}\rightarrow\big{[}M_{+},\>\operatorname{Top}/O\big{]}$$

$$M\stackrel{\hat{\iota }}{\to }\underset{\ell }{\bigvee }{S}^{n-1}\stackrel{{\pi }_j}{\to }{S}^{n-1}\stackrel{\hat{\varphi }}{\to }\mathrm{Top}/O,$$

$$\big{[}\bigvee_{\ell}S^{2n+\ell-1},Y\big{]}\rightarrow\big{[}\Sigma^{n+\ell}(M_{+}),Y].$$

$$p:{\Sigma }^{2n}(T_{+}^{\ell })\stackrel{{\kappa }^{\prime }}{\to }{\Sigma }^{n+\ell }(M_{+})\stackrel{\widehat{\iota \times 1}}{\to }\underset{\ell }{\bigvee }{S}^{2n+\ell -1}.$$

$$p^{*}:\big{[}\bigvee_{\ell}S^{2n+\ell-1},Y\big{]}\rightarrow\big{[}\Sigma^{2n}(T^{\ell}_{+}),Y\big{]}$$

$$M\stackrel{h_0^{-1}}{\to }{N}_0\stackrel{g_2}{\to }{N}_1\stackrel{h_1}{\to }M$$

$$N_0\times [0,1]\stackrel{h_0\times \text{id}}{\to }M\times [0,1]\stackrel{H}{\to }M\times [0,1]$$

$$\rho ({\alpha }^{j}c)=\{\begin{array}{ll}1& j=0\\ 0& 1\le j\le d-1.\end{array}$$

$$H_1(F_k)\simeq \mathbb{Z}\{b_1\}\oplus \mathbb{Z}F\{b_2,\dots ,b_k\}.$$

Full text

1 Introduction

Throughout this paper, $M=\operatorname{\mathbb{H}}^{n}/\pi$ denotes a closed hyperbolic manifold with fundamental group $\pi$, and $N$ denotes an exotic smooth structure (on $M$), i.e. a smooth manifold that is homeomorphic but not diffeomorphic to $M$. Define the symmetry constant of $N$ as the supremum

[TABLE]

over all Riemannian metrics $\rho$ on $N$. In this paper we study the possible values of this invariant. There is an “easy” bound

[TABLE]

that follows from Mostow rigidity and a theorem of Borel (explained below). Our main results follow:

Theorem A** (maximal symmetry constant).**

Fix $n$ such that the group $\Theta_{n}$ of exotic spheres is nontrivial. For every $d>0$, there exists a closed hyperbolic manifold $M^{n}$ and an exotic smooth structure $N$ such that $|\operatorname{Isom}(M)|\geq d$ and $s(N)=1$.

Theorem B** (arbitrarily small symmetry constant).**

Fix $n$ such that $\Theta_{n-1}\neq 0$. For every $d>1$, there exists a closed hyperbolic manifold $M^{n}$ and an exotic smooth structure $N$ such that $s(N)\leq\frac{1}{d}$.

The hypothesis $\Theta_{n}\neq 0$ is frequently true, e.g. $\Theta_{4k+3}\neq 0$ for every $k\geq 1$ and $\Theta_{4k+1}$ is nontrivial for any positive $k\notin\{1,3,7,15,31\}$. See [KM63, §7], [MS74, Appx. B], and [HHR16, Thm. 1.3].

The problem of computing $s(N)$ is related to two different problems in the study of transformation groups:

• •

Degree of symmetry. The degree of symmetry $\delta(W)$ of a manifold $W$ is defined as the largest dimension of a compact Lie group with a smooth, effective action on $W$ [HH69].

When $W=\Sigma$ is an exotic sphere, computing $\delta(\Sigma)$ is equivalent to computing the supremum

[TABLE]

over all Riemannian metrics $\rho$. Again there is a bound $\frac{1}{\dim\operatorname{SO}(n+1)}\leq s(\Sigma)\leq 1$, but the upper bound is not optimal. For example, Hsiang–Hsiang [Hsi67, HH65] prove that if $\Sigma\neq S^{n}$ has dimension $n\geq 40$, then $s(\Sigma)<\frac{n^{2}+8}{4(n^{2}+n)}<1/4$.

When $W$ is an aspherical manifold and $\pi_{1}(W)$ is centerless, then $\delta(W)=0$, i.e. $W$ does not admit a nontrivial action of a connected Lie group [Bor83]. In this case it’s fitting to define $\delta(W)$ as the largest order of a finite group that acts effectively on $W$. With this definition, for $W=N$ an exotic smooth structure on a hyperbolic manifold, $\delta(N)$ is closely related to $s(N)$; see equation (2) below.

• •

Propagating group actions [AD02]. One says that an $F$-action on $Y$ propagates across a map $f:X\rightarrow Y$ if there is an $F$-action on $X$ and an equivariant map $X\rightarrow Y$ that is homotopic to $f$. In particular, for an exotic smooth structure $N$ on a hyperbolic manifold $M$, and for a subgroup $F<\operatorname{Isom}(M)$, one can ask whether or not the action of $F$ propagates across some homeomorphism $N\rightarrow M$. This problem, and its relation to harmonic maps, is discussed in Farrell–Jones [FJ90]. Theorems A and B can be viewed as positive and negative results about propagating group actions, and give partial answers the question of [FJ90, pg. 487].

Remark. One could consider refinements of the symmetry constant such as $s_{<0}(N)=\sup_{\rho}\frac{|\operatorname{Isom}(N,\rho)|}{|\operatorname{Isom}(M)|}$, where the supremum is over all metrics with sectional curvature $K<0$. In general, $s_{<0}(N)\leq s(N)$, but computing $s_{<0}(N)$ is more difficult (e.g. it does not reduce to a Nielsen realization problem; see below). We improve upon Theorem A by giving examples for which $s_{<0}(N)=s(N)=1$.

Theorem C** (maximal symmetry, achieved by negatively-curved metric).**

Fix $n$, and assume that either $n$ is even or $|\Theta_{n}|$ is not a power of $2$. Given $d>0$, there exists a closed hyperbolic manifold $M^{n}$ and an exotic smooth structure $N$ such that $|\operatorname{Isom}(M)|\geq d$ and $N$ admits a Riemannian metric $\rho$ with negative sectional curvature so that $\operatorname{Isom}(N,\rho)\simeq\operatorname{Isom}(M)$.

If $n=4k+3$, then $|\Theta_{n}|$ is divisible by $2^{2k+1}-1$; see [MS74, Appx. B].

1.1 Techniques

The problem of determining $s(N)$ is related to a Nielsen realization problem, which will be our main point of view. By Borel [Bor83] any compact Lie group that acts effectively on $N$ is finite; furthermore, any finite subgroup of $\operatorname{Diff}(N)$ acts faithfully on $\pi=\pi_{1}(N)$. Consequently, for every $\rho$, the isometry group $\operatorname{Isom}(N,\rho)$ is a subgroup of $\operatorname{Out}(\pi)=\operatorname{Aut}(\pi)/\pi$. Furthermore, if $\dim M\geq 3$, then $\operatorname{Out}(\pi)\simeq\operatorname{Isom}(M)$ by Mostow rigidity. This explains the upper bound in (1). A subgroup $F<\operatorname{Out}(\pi)$ is said to be realized by diffeomorphisms when can we solve the lifting problem (commonly called the Nielsen realization problem — see e.g.  [BW08] and [MT18]):

[TABLE]

If $F<\operatorname{Out}(\pi)$ and $F\simeq\operatorname{Isom}(N,\rho)$ for some $\rho$, then group $F$ is a fortiori realized by diffeomorphisms. Conversely, if $F<\operatorname{Out}(\pi)$ is realized by diffeomorphisms, then by averaging a metric, we find $\rho$ with $F<\operatorname{Isom}(N,\rho)$. Therefore,

[TABLE]

where the maximum is over the subgroups $F<\operatorname{Out}(\pi)$ that are realized by diffeomorphisms. Note that $s(N)\leq\frac{|\operatorname{Im}\Psi_{N}|}{|\operatorname{Out}(\pi)|}$.

Farrell–Jones [FJ90] studied the Nielsen realization problem for $N=M\#\Sigma$, where $M^{n}$ is a closed, oriented hyperbolic manifold and $\Sigma\in\Theta_{n}$ is a nontrivial exotic sphere. The main result of [FJ90] states that if $M$ is stably parallelizable, $2\Sigma\neq 0$ in $\Theta_{n}$, and $M$ admits an orientation-reversing isometry, then $\operatorname{Im}\Psi_{N}<\operatorname{Out}(\pi)$ has index at least 2. In particular, $s(N)\leq 1/2$ for these examples.

Symmetric exotic smooth structures. Here we discuss the main components in the proof of Theorems A and C. We find our examples with $s(N)=1$ among the manifolds $N=M\#\Sigma$ studied by Farrell–Jones. Using (2), observe that $s(N)=1$ if and only if $\operatorname{Out}(\pi)$ is realized by diffeomorphisms of $N$. In particular, we must find examples where $\Psi_{N}$ is surjective. The following results refine [FJ90, Thm. 1].

Theorem 1**.**

Let $M^{n}$ be a closed, oriented hyperbolic manifold, let $\Sigma\in\Theta_{n}$ be a nontrivial exotic sphere, and let $N=M\#\Sigma$. Denote by $\operatorname{Out}^{+}(\pi)<\operatorname{Out}(\pi)$ the subgroup that acts trivially on $H_{n}(N)\simeq\mathbb{Z}$.

1. (a)

The image $\operatorname{Im}\Psi_{N}$ contains $\operatorname{Out}^{+}(\pi)$. 2. (b)

Fix $\alpha\in\operatorname{Out}(\pi)\setminus\operatorname{Out}^{+}(\pi)$. If $2\Sigma=0$ in $\Theta_{n}$, then $\alpha\in\operatorname{Im}\Psi_{N}$. The converse is true if $M$ is stably parallelizable.

Every closed hyperbolic manifold has a finite cover that is stably parallelizable [Sul79, pg. 553]. As a consequence of Theorem 1, if $2\Sigma=0$, then $\Psi_{N}$ is surjective, and if $2\Sigma\neq 0$, then $\operatorname{Im}\Psi_{N}=\operatorname{Out}^{+}(\pi)$. In any case, if $M$ does not admit an orientation-reversing isometry, then $\Psi_{N}$ is surjective. Farrell–Jones [FJ89a] show (implicitly) that reversing orientation is an obstruction to belonging to $\operatorname{Im}\Psi_{N}$ when $2\Sigma\neq 0$. According to Theorem 1, this is the only obstruction.

Having identified $\operatorname{Im}\Psi_{N}<\operatorname{Out}(\pi)$, we would like to know if this subgroup is realized by diffeomorphisms.

Theorem 2**.**

Fix $N=M\#\Sigma$ as in Theorem 1. Set $d=|\operatorname{Isom}^{+}(M)|$ and let $m\in\mathbb{N}$ be the size of the largest cyclic subgroup of $\Theta_{n}$ that contains $\Sigma$. Assume that $\gcd(d,m)$ divides $\frac{m}{|\Sigma|}$. Then $\operatorname{Out}^{+}(\pi)$ is realized by diffeomorphisms.

The assumption $\gcd(d,m)\mid\frac{m}{|\Sigma|}$ guarantees that $\Sigma\in\Theta_{n}$ has a $d$-th root. This condition is satisfied, for example, whenever $|\operatorname{Isom}^{+}(M)|$ and $|\Sigma|$ are relatively prime.

If $\operatorname{Out}^{+}(\pi)$ is realized by diffeomorphisms of $N$, then $s(N)\geq 1/2$. By Theorems 1 and 2, if $M$ is stably parallelizable and $2\Sigma\neq 0$, then $s(M\#\Sigma)$ is equal to $1/2$ or $1$, according to whether or not $M$ admits an orientation-reversing isometry. This completely solves the Nielsen realization problem in these cases.

Theorem A reduces to Theorem 2. Fixing $\Sigma\neq S^{n}$, it’s possible to find $M$ so that $|\operatorname{Isom}^{+}(M)|$ and $|\Sigma|$ are relatively prime, and $|\operatorname{Isom}^{+}(M)|$ can be made arbitrarily large. This is a consequence of a result of Belolipetsky–Lubotzky [BL05]: for any finite group $F$, there exists a closed hyperbolic $M^{n}$ with $\operatorname{Isom}(M)=F$. For their examples $\operatorname{Isom}(M)=\operatorname{Isom}^{+}(M)$. In particular, one can find examples where $\Psi_{N}:\operatorname{Diff}(N)\rightarrow\operatorname{Out}(\pi)$ is a split surjection with $|\operatorname{Out}(\pi)|$ arbitrarily large.

To prove Theorem C, one would like to promote the action of $\operatorname{Out}^{+}(\pi)$ on $N=M\#\Sigma$ produced in Theorem 2 to an action by isometries with respect to some negatively curved metric on $N$. Using a warped-metric construction of Farrell–Jones [FJ89a], it suffices to find an $M$ that is stably parallelizable, has large injectivity radius, and such that $\operatorname{Isom}^{+}(M)$ acts freely on $M$. Arranging all of these conditions simultaneously becomes delicate, especially arranging that $M$ is stably parallelizable (which is desired because it guarantees that $M\#\Sigma$ is not diffeomorphic to $M$). Because of this difficulty we take a less direct approach when $\dim M$ is odd — see Theorem 6.

Asymmetric exotic smooth structures. We explain the main ideas for proving Theorem B. For this, we consider exotic smooth structures $N=M_{c,\phi}$ obtained by removing a tubular neighborhood $S^{1}\times D^{n-1}\hookrightarrow M$ of a geodesic $c\subset M$ and gluing in $S^{1}\times D^{n-1}$ by a diffeomorphism $\mathbbm{1}\times\phi$ of $S^{1}\times S^{n-2}$, where $\phi\in\operatorname{Diff}(S^{n-2})$ is not isotopic to the identity. Farrell–Jones [FJ93] prove that $M_{c,\phi}$ is often an exotic smooth structure on $M$.

The strategy for proving Theorem B is to find $N=M_{c,\phi}$ and $F\simeq\mathbb{Z}/d\mathbb{Z}$ in $\operatorname{Out}(\pi)$ so that $\operatorname{Im}\Psi_{N}\cap F=1$. This condition implies that the index of $\operatorname{Im}\Psi_{N}<\operatorname{Out}(\pi)$ is at least $|F|$, so $s(N)\leq\frac{1}{|F|}$. To show $F\cap\operatorname{Im}\Psi_{N}=1$, we study how the smooth structure on $M_{c,\phi}$ changes if we choose a different geodesic $c$. This is complementary to [FJ93, Thm. 1.1], which studies how the smooth structure changes when the geodesic is fixed and the isotopy class $[\phi]\in\pi_{0}\operatorname{Diff}(S^{n-2})\simeq\Theta_{n-1}$ is changed. In Theorem 8 we give a criterion to guarantees that $M_{c_{1},\phi}$ and $M_{c_{2},\phi}$ are not concordant, i.e. there is no smooth structure on $M\times[0,1]$ that restricts to $M_{c_{1},\phi}\sqcup M_{c_{2},\phi}$ on the boundary. This is one of the main technical ingredients in the proof of Theorem B.

The proof of Theorem B works equally well when $M$ is nonuniform, but we won’t discuss this further.

Theorem B proves that $s(N)$ may be arbitrarily close to 0, as $N$ varies over exotic smooth structures on all hyperbolic $n$-manifolds (when $\Theta_{n-1}\neq 0$), but if we fix the homeomorphism type, we know that $s(N)\geq\frac{1}{|\operatorname{Isom}(M)|}$. It would be interesting to know if there are examples where this lower bound is achieved. Of course if $\operatorname{Isom}(M)=1$, then $s(N)=1=\frac{1}{|\operatorname{Isom}(M)|}$, so to make this interesting one should ask for examples such that $\operatorname{Isom}(M)$ is large.

Question 3**.**

Does there exist $n$ so that for every $d>0$, there exists a hyperbolic manifold $M^{n}$ and an exotic smooth structure $N$ such that $|\operatorname{Isom}(M)|\geq d$ and $s(N)=\frac{1}{|\operatorname{Isom}(M)|}$?

Note that $s(N)=\frac{1}{|\operatorname{Isom}(M)|}$ if and only if $\Psi_{N}:\operatorname{Diff}(N)\rightarrow\operatorname{Out}(\pi)$ is trivial. Equivalently, $\operatorname{Isom}(N,\rho)=1$ for every Riemannian metric $\rho$.

Section outline. In §2 we prove Theorems 1 and 2 and discuss some related questions of interest. In §3 we discuss the work of Belolipetsky–Lubotzky and use it to prove Theorem C. Finally, in §4 we prove Theorem B; specifically, we study when two smooth structures $M_{c_{1},\phi}$ and $M_{c_{2},\phi}$ are concordant, which we use as an obstruction to Nielsen realization.

Acknowledgements. The authors would like to thank I. Belegradek and S. Cappell for helpful and interesting conversations. M.B. has been supported by the Special Priority Program SPP 2026 “Geometry at Infinity” funded by the Deutsche Forschungsgemeinschaft (DFG).

2 Symmetry constant for $N=M\#\Sigma$

In this section we prove Theorems 1 and 2.

2.1 The image of $\Psi_{N}:\operatorname{Diff}(N)\rightarrow\operatorname{Out}(\pi)$

Proof of Theorem 1.

Let $N=M\#\Sigma$ as in the theorem. It will be convenient to fix $p\in M$ and a small metric ball $B=B_{r}(p)$ where the connected sum is performed.

First we prove (a). For this we fix $\alpha\in\operatorname{Out}^{+}(\pi)\simeq\operatorname{Isom}^{+}(M)$ and define $f\in\operatorname{Diff}(N)$ so that $\Psi_{N}(f)=\alpha$. View $\alpha$ as an isometry of $M$, and choose an isotopy $\alpha_{t}\in\operatorname{Diff}(M)$ so that $\alpha_{0}=\alpha$ and $\alpha_{1}(B)=B$ and \alpha_{1}\bigr{|}_{B}\in O(n) is an isometry of the ball; for example, if the radius $r$ is sufficiently small, then we can isotope $\alpha(B)$ to $B$ in $M$ through isometric embeddings, and then extend the isotopy of $B$ to an ambient isotopy. Since $\alpha$ is orientation-preserving, \alpha_{1}\bigr{|}_{B} belongs to the identity component $SO(n)\subset O(n)$, and it is easy to see then that $\alpha_{1}$ induces a diffeomorphism $f:N\rightarrow N$; for example, isotope \alpha_{1}\bigr{|}_{B} further so that \alpha_{1}\bigr{|}_{B_{r/2}(p)} is the identity and perform the connected sum along $B_{r/2}(p)$ instead of $B_{r}(p)$. This proves part (1).

To prove (b), assume that $\alpha\in\operatorname{Out}(\pi)\setminus\operatorname{Out}^{+}(\pi)$. Viewing $\alpha$ as an orientation-reversing isometry of $M$, the argument above defines an orientation-reversing diffeomorphism $h:M\#\Sigma\rightarrow M\#\overline{\Sigma}$ that induces $\alpha$ (recall that for $A\#B$, if the identification of the attaching disk is changed by an orientation-reversing involution, then the result is $A\#\overline{B}$, where $\overline{B}$ is $B$ with the opposite orientation). If $2\Sigma=0$ in $\Theta_{n}$, then $\Sigma=\overline{\Sigma}$ (because $\overline{\Sigma}=-\Sigma$ in $\Theta_{n}$), so $h\in\operatorname{Diff}(N)$ and $\Psi_{N}(h)=\alpha$. This proves the first statement of (b). The converse is already to contained in [FJ90, Thm. 1]. In short, if $\Psi_{N}(f)=\alpha$ for some $f\in\operatorname{Diff}(N)$, then $h\circ f$ is an orientation-preserving diffeomorphism $M\#\Sigma\rightarrow M\#\overline{\Sigma}$. When $M$ is stably parallelizable, this implies that $2\Sigma=0$ by [FJ89a, §2]. ∎

2.2 Sections of $\Psi_{N}:\operatorname{Diff}(N)\rightarrow\operatorname{Im}\Psi$

Proof of Theorem 2.

Since $M$ is hyperbolic, $\operatorname{Out}(\pi)$ is realized by isometries of $M$ (by Mostow rigidity). Set $F=\operatorname{Isom}^{+}(M)$. Since $F$ is finite, there exists $p\in M$ whose stabilizer in $F$ is trivial. Choose a ball $B$ around $p$ whose $F$-translates are disjoint. By assumption, $\gcd(|F|,m)$ divides $\frac{m}{|\Sigma|}$, which implies that there exists $\Sigma^{\prime}\in\Theta_{n}$ so that $\Sigma=|F|\cdot\Sigma^{\prime}$. Then $N=M\#\Sigma$ is diffeomorphic to $M\#\Sigma^{\prime}\#\cdots\#\Sigma^{\prime}$, where $\Sigma^{\prime}$ appears $|F|$ times. If we form the connected sum along the union of balls $F.B$, then we can extend the action of $F$ on $M\setminus F.B$ to a smooth $F$-action on $N=M\#\Sigma^{\prime}\#\cdots\#\Sigma^{\prime}$ by rigidly permuting the exotic spheres. ∎

Remark. One might think that the above argument could be used to define an action of $\operatorname{Out}(\pi)$ on $N$ under a similar constraint on $|\operatorname{Out}(\pi)|$ and $|\Sigma|$. This would contradict the fact that $\Psi_{N}$ is frequently not surjective when $M$ admits an orientation-reversing isometry. In the argument above, when $M$ admits an orientation-reversing isometry, one obtains an action of $\operatorname{Out}(\pi)$ on $M\#k\Sigma^{\prime}\#k\overline{\Sigma^{\prime}}$, where $k=|\operatorname{Out}(\pi)|/2$. But $M\#k\Sigma^{\prime}\#k\overline{\Sigma^{\prime}}$ is diffeomorphic to $M$, not $N$.

It would be interesting to know if $\operatorname{Out}^{+}(\pi)$ ever acts on $N=M\#\Sigma$ when $N$ has no “obvious” symmetry:

Question 4**.**

Is Theorem 2 ever true without the assumption $\gcd(d,m)\mid\frac{m}{|\Sigma|}$? For example, fix $\alpha\in\operatorname{Isom}^{+}(M)$ of order $d$, and assume that $\alpha$ acts freely. Choose $\Sigma\in\Theta_{n}$ that does not admit a $d$-th root. Prove or disprove that the subgroup $\langle\alpha\rangle\simeq\mathbb{Z}/d\mathbb{Z}$ in $\operatorname{Out}^{+}(\pi)$ is realized by diffeomorphisms of $N=M\#\Sigma$.

In this direction, it would be interesting to know how the choice of $\Sigma$ affects the answer to Question 4. For instance, in the study of the symmetry constant of $\Sigma\in\Theta_{n}$, there is a marked difference between (1) the standard sphere $\Sigma=S^{n}$, (2) the nontrivial exotic spheres that bound a parallelizable manifold $\Sigma\in bP_{n+1}\setminus\{S^{n}\}$, and (3) the remaining exotic spheres $\Sigma\in\Theta_{n}\setminus bP_{n+1}$. See [HH69]. Does this distinction play a role in Question 4?

Note that the subtlety in Question 4 disappears in the topological category: if $W$ is an aspherical manifold with $\pi_{1}(W)\simeq\pi$, then $\operatorname{Homeo}(W)\rightarrow\operatorname{Out}(\pi)$ is a split surjection because $W$ and $M$ are homeomorphic by the solution of Farrell–Jones to the Borel conjecture in this case; see [Far02, Cor. 3 in §5].

We mention another problem related to Question 4. For this, let $W^{n}$ be an exotic smooth structure on the torus $T^{n}$. There is a surjective homomorphism $\operatorname{Diff}^{+}(W)\rightarrow\operatorname{Out}^{+}(\pi_{1}(W))\simeq\operatorname{SL}_{n}(\mathbb{Z})$, and whether or not this homomorphism splits is unknown. One approach to this question is to focus on maximal abelian subgroups of $\operatorname{SL}_{n}(\mathbb{Z})$ and try to use the dynamics of Anosov diffeomorphisms; see [FKS13, Question 1.4] and also [BRHW17]. Alternatively, an obstruction to realizing finite subgroups $F<\operatorname{SL}_{n}(\mathbb{Z})$ as in Question 4 could provide an approach to the splitting problem for certain $W=T^{n}\#\Sigma$.

3 Realization by isometries

In this section, we prove Theorem C. The starting point of our argument is the following result from [BL05, Thm. 1.1 and §6.3].

Theorem 5** (Belolipetsky–Lubotzky).**

For every $n\geq 2$ and every finite group $F$, there exists infinitely many compact $n$-dimensional hyperbolic manifolds $M$ with $\operatorname{Isom}(M)=\operatorname{Isom}^{+}(M)\simeq F$.

The main result we prove here is as follows.

Theorem 6**.**

Fix a finite group $F$ and fix $R>0$. Among the hyperbolic manifolds $M^{n}$ with $\operatorname{Isom}(M)=\operatorname{Isom}^{+}(M)\simeq F$, there exists $M$ such that

1. (a)

the group $F$ acts freely on $M$, 2. (b)

there is a cover $\widehat{M}\rightarrow M$ of degree $\ell\in\{1,2,4\}$ so that $\widehat{M}$ is stably parallelizable, and 3. (c)

$\operatorname{InjRad}(M)>R$.

Furthermore, for (b), if $n$ is even, then we can take $\ell=1$.

Next we deduce Theorem C from Theorem 6.

Proof of Theorem C.

Fix $d>0$. If $n$ is even, take any nontrivial $\Sigma\in\Theta_{n}$ and let $F$ be a group with $|F|\geq d$ and $\gcd(|F|,|\Sigma|)=1$. If $|\Theta_{n}|\neq 2^{i}$, take $\Sigma\in\Theta_{n}$ nontrivial of odd order and let $F$ be a $2$-primary group with $|F|\geq d$. In either case, there exists $\Sigma^{\prime}\in\Theta_{n}$ with $\Sigma=|F|\cdot\Sigma^{\prime}$. By Belolipetsky–Lubotzky and Theorem 2, for every $M$ with $\operatorname{Isom}(M)\simeq\operatorname{Isom}^{+}(M)\simeq F$, the group $F$ acts by diffeomorphisms of $N=M\#\Sigma\simeq M\#\Sigma^{\prime}\#\cdots\#\Sigma^{\prime}$. We need to show we can choose $M$ and a negatively-curved metric $\rho$ on $N$ so that $F=\operatorname{Isom}(N,\rho)$ in $\operatorname{Diff}(N)$.

According to [FJ89a, Prop. 1.3], there is a constant $\tau_{n}>0$ so that if $M^{n}$ has injectivity radius $\operatorname{InjRad}(M)>\tau_{n}$, then $N=M\#\Sigma$ admits a negatively curved metric. This metric agrees with the hyperbolic metric on $M$ away from the disk where the connected sum is performed, and on that disk, the metric is radially symmetric. Choose $M$ satisfying Theorem 6 with $R=|F|\cdot\tau_{n}$ and such that $F$ acts freely on $M$, so the quotient $\overline{M}=M/F$ is a hyperbolic manifold. Furthermore,

[TABLE]

We prove this below. Now fix $r$ with $\tau_{n}<r<\operatorname{InjRad}(\overline{M})$. From (3) it follows that for any ball $B=B_{r}(p)$ in $M$, the $F$-translates of $B$ are disjoint. Fix such a ball $B$. As in the proof of Theorem 2, write $\Sigma=|F|\cdot\Sigma^{\prime}$ and consider $M_{0}=M\setminus F.B$. The manifold $N$ is obtained by gluing $\mathbb{D}^{n}$ to each boundary component of $M_{0}$ by a fixed diffeomorphism $f\in\operatorname{Diff}(S^{n-1})$. Using the technique in [FJ89a], we give $N$ a Riemannian metric $\rho$ that agrees with the hyperbolic metric on $M_{0}$ and is a warped-product metric on each $\mathbb{D}^{n}$. Since $r>\tau_{n}$, [FJ89a, §3] guarantees that the resulting metric has negative curvature. The group $F$ acts on $N$ as in Theorem 2, and by construction it acts by isometries for the metric $\rho$.

Now we explain the inequality (3). To see the first inequality, note that $2\operatorname{InjRad}(M)=\operatorname{sys}(M)$, where $\operatorname{sys}(M)$ is systole, i.e. the length of the shortest geodesic. Under a $d$-fold isometric cover $M\rightarrow\overline{M}$, if $\overline{\gamma}$ is a closed geodesic of $\overline{M}$ and $\gamma\subset M$ is a connected component of its preimage, then $\text{length}(\gamma)\leq d\cdot\text{length}(\overline{\gamma})$. It follows that $\operatorname{sys}(M)\leq d\cdot\operatorname{sys}(\overline{M})$.

It remains is to show that $N$ is not diffeomorphic to $M$. When $n$ is even, then by Theorem 6 we can assume that $M$ is stably parallelizable and so $M$ is not diffeomorphic to $M\#\Sigma$ by Farrell–Jones [FJ89a]. In the general case, $M$ has a stably parallelizable cover of degree 2 or 4. Suppose for a contradiction that $M\#\Sigma$ is diffeomorphic to $M$. Lifting to the cover $\widehat{M}\rightarrow M$, we find that $\widehat{M}\#\ell\Sigma$ is diffeomorphic to $\widehat{M}$. Note that $\ell\Sigma\neq 0$ in $\Theta_{n}$ since $\Sigma$ has odd order and $\ell\in\{2,4\}$. Since $\widehat{M}$ is stably parallelizable, by [FJ89a, Prop. 1.2], we conclude that $\widehat{M}\#\ell\Sigma$ is not diffeomorphic to $\widehat{M}$. This is a contradiction, so $N$ is not diffeomorphic to $M$ as desired. This completes the proof. ∎

Next we prove Theorem 6. Fix a finite group $F$. In what follows $M=\operatorname{\mathbb{H}}^{n}/\pi$ will always denote one of the Belolipetsky–Lubotzky manifolds with $\operatorname{Isom}(M)=\operatorname{Isom}^{+}(M)\simeq F$. We have to explain why $M$ can be chosen to satisfy (a), (b), and (c). We will see that [BL05, Thm. 2.1] already shows that (a) can be arranged, and that (b) can be arranged by modifying the proof of [BL05, Prop. 2.2]. Part (c) requires a different, separate argument. All of these arguments involve passing to certain congruence covers, so once we explain why (a), (b), and (c) can be arranged individually, it will be evident that they can be arranged simultaneously.

Recollection of Belolipetsky–Lubotzky [BL05]. Here we summarize the main results of [BL05], especially the aspects needed for our proof. Let $\Gamma$ be a finitely generated group. Assume that $\Delta\lhd\Gamma$ is finite-index, normal, and that $\Delta$ surjects to a finite-rank free group:

[TABLE]

for some $r\geq 2$. The conjugation action of $N_{\Gamma}(K)$ on $\Delta$ preserves $K$, so $N_{\Gamma}(K)$ acts on $F_{r}$ by automorphisms. Let $D<N_{\Gamma}(K)$ be the subgroup that acts on $F_{r}$ by inner automorphisms. With this setup, the main algebraic construction of [BL05, Thm. 2.1] asserts that for any finite group $F$, there exists a finite-index subgroup $\pi<D$ with $N_{\Gamma}(\pi)/\pi\simeq F$ (in their notation, they use $M$ instead of $K$ and $B$ instead of $\pi$).

In the application to hyperbolic manifolds, define $\Gamma$ as the commensurator $\operatorname{Comm}(\Lambda)$ of a Gromov–Piatetski-Shapiro [GPS88] non-arithmetic lattice $\Lambda<\operatorname{SO}(n,1)$. By work of Mostow and Margulis, $\operatorname{Comm}(\Lambda)$ is a maximal discrete subgroup of $\operatorname{Isom}(\operatorname{\mathbb{H}}^{n})$, so for any $\pi<\Gamma$,

[TABLE]

Hence to find $M=\operatorname{\mathbb{H}}^{n}/\pi$ with $\operatorname{Isom}(M)\simeq F$, it suffices to find $\pi<\Gamma$ with $N_{\Gamma}(\pi)/\pi\simeq F$.

To define $\Delta$, denote $G=O(n,1)$ and let $\mathcal{O}_{S}$ be ring of definition of $\Gamma$, so $\Gamma<G(\mathcal{O}_{S})$. Let $\mathfrak{p}\subset\mathcal{O}_{S}$ be a prime ideal and denote $p\in\mathbb{N}$ the prime with $(p)=\mathfrak{p}\cap\mathbb{Z}$. We only deal with prime ideals $\mathfrak{p}$ where $\mathcal{O}_{S}/\mathfrak{p}\simeq\mathbb{F}_{p}$. Equivalently, $p$ splits completely in $\mathcal{O}_{S}$; there are infinitely many such $\mathfrak{p}$ by Chebotarev’s theorem. Reduction mod $\mathfrak{p}$ defines a map $\alpha_{\mathfrak{p}}:\Gamma\rightarrow G(\mathcal{O}_{S}/\mathfrak{p})\simeq O_{n+1}(p)$ to an orthogonal group over $\mathbb{F}_{p}$. Define $\Gamma(\mathfrak{p})=\ker(\overline{\alpha}_{\mathfrak{p}})$, where $\overline{\alpha}_{\mathfrak{p}}:\Gamma\rightarrow O_{n+1}(p)\rightarrow\operatorname{PO}_{n+1}(p)$. The group $\Delta$ is defined as $\Lambda\cap\Gamma(\mathfrak{p})$.

To ensure $\Delta\lhd\Gamma$, we want $\Lambda\lhd\Gamma$. In order to arrange this, after we’ve defined $\Gamma$, we replace $\Lambda$ with a finite-index subgroup (still denoted $\Lambda$) so that $\Lambda\lhd\Gamma$ (note that this replacement does not change $\operatorname{Comm}(\Lambda)$). The group $\Delta$ surjects to a free group: By the cut-and-paste nature of the construction of [GPS88], $\Lambda$ is either an amalgamated product or an HNN extension. For definiteness assume $\Lambda=\Lambda_{1}*_{\Lambda_{3}}\Lambda_{2}$. Denoting $\Omega_{n+1}(p)=[O_{n+1}(p),O_{n+1}(p)]$, by strong approximation, for all but finitely many $\mathfrak{p}$, the image of $\overline{\alpha}_{\mathfrak{p}}:\Lambda\rightarrow\operatorname{PO}_{n+1}(p)$ contains $Q_{p}:=\operatorname{P\Omega}_{n+1}(p)$, and the same is true for the restriction to $\Lambda_{1},\Lambda_{2}$. Without loss of generality, we may assume $\operatorname{Im}(\overline{\alpha}_{\mathfrak{p}})=Q_{p}$ (replace $\Lambda$ by the intersection of all index-2 subgroups of $\Lambda$). Denoting $T_{p}=\overline{\alpha}_{p}(\Lambda_{3})$, the map $\overline{\alpha}_{p}$ factors through surjective maps $\Lambda\xrightarrow{s}Q_{p}*_{T_{p}}Q_{p}\xrightarrow{t}Q_{p}$. Then $\Delta=\ker(t\circ s)$ surjects onto $\ker t$, which is a free group of rank $r\geq 2$ [BL05, Prop. 3.4].

Proof of Theorem 6. Fix a finite group $F$. We use the setup of the proceeding paragraphs. In particular, $\pi<D$ will always denote a subgroup with $N_{\Gamma}(\pi)/\pi\simeq F$, and our aim is to show that $\pi$ can be chosen in such a way that $M=\operatorname{\mathbb{H}}^{n}/\pi$ has properties (a), (b), and (c).

Part (a). By [BL05, pg. 465] the group $N_{\Gamma}(\pi)$ is contained in D=\ker\big{[}N_{\Gamma}(K)\rightarrow\operatorname{Out}(F_{r})\big{]}, and [BL05, §5] shows that $D$ is contained in $\Gamma(\mathfrak{p})$, which is torsion-free for $p$ large. It follows that $\operatorname{Isom}(M)\simeq N_{\Gamma}(\pi)/\pi$ acts freely on $M$: if $x\in M$ is fixed by $g\neq 1\in\operatorname{Isom}(M)$, then $g$ lifts to $\tilde{g}\in N_{\Gamma}(\pi)$ that acts on $\operatorname{\mathbb{H}}^{n}$ with a fixed point, but this contradicts the fact that $N_{\Gamma}(\pi)$ is torsion-free.

Part (b). As mentioned in part (a), we can arrange that $\pi<\Gamma(\mathfrak{p})$. Our main task for part (b) will be to show that we can also arrange that $\pi<\Gamma(\mathfrak{p})\cap\Gamma(\mathfrak{q})$, where $\mathfrak{p},\mathfrak{q}\subset\mathcal{O}_{S}$ are prime ideas with $\mathcal{O}_{S}/\mathfrak{p}\simeq\mathbb{F}_{p}$ and $\mathcal{O}_{S}/\mathfrak{q}\simeq\mathbb{F}_{q}$ for distinct primes $p,q$. Before we do this, we explain why this is enough to conclude that $M=\operatorname{\mathbb{H}}^{n}/\pi$ has the desired stably parallelizable cover.

Suppose that $M=\operatorname{\mathbb{H}}^{n}/\pi$ with $\pi<\Gamma(\mathfrak{p})\cap\Gamma(\mathfrak{q})$. We will show that there is a cover $\widehat{M}\rightarrow M$ of degree 1, 2, or 4 so that $\widehat{M}$ has a tangential map $\widehat{M}\rightarrow S^{n}$, and hence $\widehat{M}$ is stably parallelizable. The group $\pi$ is a subgroup of the identity component $\operatorname{SO}_{0}(n,1)<\operatorname{SO}(n,1)$. The inclusions $\pi\hookrightarrow\operatorname{SO}_{0}(n,1)\hookrightarrow\operatorname{SO}_{n+1}(\mathbb{C})$ define flat bundles over $M$. By Deligne–Sullivan [DS75], there is a particular cover $\widehat{M}\rightarrow M$ so that the map $\widehat{M}\rightarrow M\rightarrow B\operatorname{SO}_{n+1}(\mathbb{C})$ is homotopically trivial. This cover is the one corresponding to the subgroup $\widehat{\pi}=\pi\cap\ker(\alpha_{\mathfrak{p}})\cap\ker(\alpha_{\mathfrak{q}})$ of $\pi$. Note that the index $[\pi:\widehat{\pi}]$ is 1, 2, or 4 because $\ker(\alpha_{\mathfrak{p}})$ has index 2 in $\ker(\overline{\alpha}_{\mathfrak{p}})$. Furthermore, if $n$ is even, then $\operatorname{SO}_{n+1}(p)<O_{n+1}(p)$ has trivial center, so $\operatorname{SO}_{n+1}(p)\simeq\operatorname{PSO}_{n+1}(p)$, which implies that $\widehat{\pi}=\pi$.

Since there is a fibration

[TABLE]

and $\widehat{M}\rightarrow B\operatorname{SO}_{0}(n,1)\rightarrow B\operatorname{SO}_{n+1}(\mathbb{C})$ is trivial, the map $\widehat{M}\rightarrow B\operatorname{SO}_{0}(n,1)$ lifts to $\operatorname{SO}_{n+1}(\mathbb{C})/\operatorname{SO}_{0}(n,1)$, which is homotopy equivalent to $\operatorname{SO}(n+1)/\operatorname{SO}(n)\simeq S^{n}$. This map $\widehat{M}\rightarrow S^{n}$ is a tangential map by Okun [Oku01, §5]. This completes the construction of the stably parallelizable cover.

Now we show we can find $M$ with isometry group $F$ and fundamental group $\pi<\Gamma(\mathfrak{p})\cap\Gamma(\mathfrak{q})$. As above, fix $\mathfrak{p}\subset\mathcal{O}_{S}$ such that $\alpha_{p}:\Lambda\rightarrow Q_{p}$ is surjective and also $\alpha(\Lambda_{1})=\alpha(\Lambda_{2})=Q_{p}$.

Observation. Fix a prime ideal $\mathfrak{q}\subset\mathcal{O}_{S}$ and denote $q\in\mathbb{N}$ the prime with $(q)=\mathfrak{q}\cap\mathbb{Z}$. If the image of $\overline{\alpha}_{\mathfrak{q}}:\Lambda(p)\rightarrow\operatorname{PO}_{n+1}(q)$ contains $Q_{q}$, then the image of $\overline{\alpha}_{\mathfrak{p},\mathfrak{q}}:\Lambda\to\operatorname{PO}_{n+1}(p)\times\operatorname{PO}_{n+1}(q)$ defined by

[TABLE]

contains $Q_{p}\times Q_{q}$. Indeed, if $(x,y)\in Q:=Q_{p}\times Q_{q}$, then one has that $\overline{\alpha}_{\mathfrak{p}}(g)=x$ for some $g\in\Lambda$ and also $\overline{\alpha}_{\mathfrak{q}}(h)=\overline{\alpha}_{\mathfrak{q}}(g)^{-1}y$ for some $h\in\Lambda(\mathfrak{p})$. Thus $\overline{\alpha}_{\mathfrak{p},\mathfrak{q}}(gh)=(x,y)$.

We use the observation together with the strong approximation theorem to conclude that for all but finitely many of the infinitely many primes $q$ that split completely, the image of each of $\Lambda$, $\Lambda_{1}$, and $\Lambda_{2}$ in $\operatorname{PO}_{n+1}(p)\times\operatorname{PO}_{n+1}(q)$ contains $Q_{p}\times Q_{q}$. As before, we may assume (by replacing $\Lambda$ with a finite-index subgroup) that $\overline{\alpha}_{\mathfrak{p},\mathfrak{q}}(\Lambda)=Q_{p}\times Q_{q}$.

Set $T=\overline{\alpha}_{\mathfrak{p},\mathfrak{q}}(\Lambda_{3})$. The subgroup $T<Q$ has the property that there are no nontrivial $N\lhd Q$ such that $1\leq N\leq T$ (compare [BL05, §3.2]). This holds essentially for the same reasons it holds for $T_{p}<Q_{p}$ (see [BL05, §5]). In our case, we only need to notice that $T\leq\operatorname{PO}_{n}(p)\times\operatorname{PO}_{n}(q)$, while the only nontrivial proper normal subgroups of $Q$ are $Q_{p}\times 1$ and $1\times Q_{q}$ (the latter fact holds because $Q_{p}$ and $Q_{q}$ are simple if $p,q$ are sufficiently large and $Q_{p}\not\simeq Q_{q}$).

Setting $\Delta=\ker(\overline{\alpha}_{\mathfrak{p},\mathfrak{q}})=\Lambda\cap\Gamma(\mathfrak{p})\cap\Gamma(\mathfrak{q})$, we may repeat the argument of [BL05, §5] to conclude that $\pi<D$ is contained in $\Gamma(\mathfrak{p})\cap\Gamma(\mathfrak{q})$. This finishes part (b).

Part (c). We explain why we can arrange for $M$ to have isometry group $F$ and arbitrarily large injectivity radius. This will follow (using Proposition 7 below) from the fact that $\pi$ is a subgroup of matrices $\operatorname{SL}_{m}(\mathcal{O}_{S})$ with coefficients in the ring $\mathcal{O}_{S}$ of $S$-integers in a number field $L$. Before proving Proposition 7 we recall a few facts about $\mathcal{O}_{S}$. Here $\mathcal{O}$ is the ring of integers in $L$, and $S$ is a finite set of places (i.e. an equivalence class of absolute value on $L$) that includes all of the Archimedean places, and $\mathcal{O}_{S}=\{x\in L:t(x)\leq 1\text{ for all places }t\notin S\}$.

For our proof of Proposition 7, we recall the description of the set of all places of $L$. This is the content of Ostrowski’s theorem [Jan96, Ch. II]. The Archimedean places all come from embeddings of $L$ into $\mathbb{R}$ or $\mathbb{C}$. The non-Archimedean places come from prime ideals $\mathfrak{q}\subset\mathcal{O}$ as follows. Given $\mathfrak{q}$, for $a\in\mathcal{O}$ define $\nu_{\mathfrak{q}}(a)\in\mathbb{Z}_{\geq 0}$ as the multiplicity of $\mathfrak{q}$ appearing in the prime factorization of the ideal $(a)\subset\mathcal{O}$; this is extended to $x=\frac{a}{b}\in L$ by $\nu_{\mathfrak{q}}(x)=\nu_{\mathfrak{q}}(a)-\nu_{\mathfrak{q}}(b)$. Denoting the norm $N(\mathfrak{q})=|\mathcal{O}/\mathfrak{q}|$, the function $t_{\mathfrak{q}}(x)=N(\mathfrak{q})^{-\nu_{\mathfrak{q}}(x)}$ defines a place of $L$. The set of all places (normalized in the way we have described) satisfies the product formula $\prod t(x)=1$ for any $x\in L^{\times}$ [Jan96, Ch. II, §6]. For future reference, observe that if $a\in\mathcal{O}$ and $\mathfrak{q}\nmid a$, then $t_{\mathfrak{q}}(a)=1$, so only finitely many terms in the product $\prod t(x)$ differ from 1. Note also that if $(a)=\mathfrak{q}_{1}^{n_{1}}\cdots\mathfrak{q}_{f}^{n_{f}}$ is the prime factorization, then $N(a)=N(\mathfrak{q}_{1})^{n_{1}}\cdots N(\mathfrak{q}_{f})^{n_{f}}$, so by the product formula, $N(a)$ is also equal to the product $\prod_{t\mid\infty}t(a)$ over Archimedean places of $L$.

Proposition 7** (Injectivity radius growth in congruence covers).**

Let $V$ be a closed aspherical Riemannian manifold with fundamental group $\pi$. Suppose there exists an injection $\pi\hookrightarrow\operatorname{SL}_{m}(\mathcal{O}_{S})$, where $\mathcal{O}_{S}$ is the ring of $S$-integers in a number field $L$. For an ideal $\mathfrak{k}\subset\mathcal{O}$, denote

[TABLE]

and let $V_{\mathfrak{k}}$ be the cover of $V$ with fundamental group $\pi(\mathfrak{k}):=\pi\cap\operatorname{SL}_{m}(\mathfrak{k})$. Then there are constants $C,D$ (depending only on $V$, $m$, and $K$, but not $\mathfrak{k}$) so that $\operatorname{InjRad}(V_{\mathfrak{k}})\geq C\log k+D$, where $(k)=\mathfrak{k}\cap\mathbb{Z}$.

This statement is similar to the “Elementary Lemma” of [Gro96, §3.C.6]. The proof below is based on, and has some overlap with, the argument in [GL14, §4].

Proof of Proposition 7.

Let $\widetilde{V}$ be the universal cover of $V$.

Fix the ideal $\mathfrak{k}$, and set $R=\operatorname{InjRad}(V_{\mathfrak{k}})$. By definition of $\operatorname{InjRad}$, there exists $y,z\in\widetilde{V}$ and $\eta\in\pi(\mathfrak{k})$ so that $y,\eta y$ are both contained in the ball $B_{2R}(z)$. Then $d(y,\eta y)\leq 4R$; equivalently

[TABLE]

To prove the proposition, we will give a lower bound on $d(y,\eta y)$.

Since $V$ is compact, $\pi$ is finitely generated. Consider the generating set associated to the Dirichlet fundamental domain $\mathcal{D}$ centered at $y$ for the action of $\pi$ on $\widetilde{V}$ (generators are those $g\in\pi$ for which $g(\mathcal{D})\cap\mathcal{D}\neq\operatorname{\varnothing}$). For the word length $w:\pi\rightarrow\mathbb{Z}_{\geq 0}$ associated to this generating set, there is a bound w(\eta)\leq c_{1}\cdot\big{[}d(y,\eta y)+1\big{]}, obtained as follows. Take a geodesic $\gamma$ connecting $y,\eta y$, and cover it by $\lfloor d(y,\eta y)\rfloor+1$ balls of radius 1. There is $c_{1}>0$ so that each ball intersects at most $c_{1}$ translates of $\mathcal{D}$, so $\gamma$ intersects at most c_{1}\cdot\big{[}d(y,\eta y)+1\big{]} translates of $\mathcal{D}$. This proves the aforementioned bound, which is equivalent to

[TABLE]

To finish the proof, we prove

[TABLE]

for some constants $c_{2},c_{3}$. Now we use the assumptions that $\pi<\operatorname{SL}_{m}(\mathcal{O}_{S})$ and $\eta\in\operatorname{SL}_{m}(\mathfrak{k})$. For $X=(x_{ij})\in\operatorname{SL}_{m}(L)$ and $s\in S$, define

[TABLE]

By the formula for matrix multiplication $|XY|_{S}\leq m\>|X|_{S}|Y|_{S}$. Write $\eta=X_{1}\cdots X_{w(\eta)}$ with $X_{i}\in\operatorname{SL}_{m}(\mathcal{O}_{S})$ belonging to our chosen generating set of $\pi$. Then $|\eta|_{S}\leq m^{w(\eta)-1}\cdot M^{w(\eta)}$, where $M$ is the maximum value of $|\cdot|_{S}$ on generators of $\pi$. On the other hand, we will show that $|\eta|_{S}\geq\ell\cdot k^{1/\ell}-\ell$, where $(k)=\mathfrak{k}\cap\mathbb{Z}$ and $\ell=|S|$. Then altogether we have

[TABLE]

which gives a bound as in (4) after taking log. Note that $\log(k^{1/\ell}-1)=\log(k^{1/\ell})+\log(1-k^{-1/\ell})$ and $\log(1-k^{-1/\ell})$ is bounded below by the constant $\log(1-2^{-1/\ell})$.

Now we prove $|\eta|_{S}\geq\ell\cdot k^{1/\ell}-\ell$. Since $\eta\neq\operatorname{Id}$, some entry $\eta_{ij}$ has the form $1+x$ or $x$, where $x\in\mathfrak{k}\mathcal{O}_{S}$ is nonzero. Write $x=\frac{a}{b}\cdot x_{1}$, where $x_{1}\in\mathfrak{k}$ and the only primes dividing $a,b$ are primes in $S$. By the product formula

[TABLE]

Furthermore, $N(x_{1})\geq N(\mathfrak{k})\geq k$ because $(x_{1})\subset\mathfrak{k}$ and $\mathbb{Z}/k\mathbb{Z}\subset\mathcal{O}/\mathfrak{k}$. Therefore, $\prod_{s\in S}s(x)\geq k$.

Next we show that $\prod_{s\in S}s(x)\geq k$ implies that $|x|_{S}:=\sum_{s\in S}s(x)\geq\ell k^{1/\ell}$. This follows from some calculus: we want to minimize the function $\phi(x_{1},\ldots,x_{\ell})=x_{1}+\cdots+x_{\ell}$ under the constraint $x_{1}\cdots x_{\ell}\geq k$. Since $\phi$ has no critical points, the minimum is achieved on the set $x_{1}\cdots x_{\ell}=k$. Using Lagrange multipliers, one finds that $\phi$ has a unique minimum at $x=(k^{1/\ell},\ldots,k^{1/\ell})$ and the minimum value is $\phi(x)=\ell\cdot k^{1/\ell}$.

Since $\eta_{ij}$ is either $x$ or $1+x$, in either case $|\eta_{ij}|_{S}\geq\sum_{s\in S}[s(x)-1]\geq\ell\cdot k^{1/\ell}-\ell$. Combining everything we conclude that

[TABLE]

This completes the proof. ∎

4 Symmetry constant for $N=M_{c,\phi}$

In this section we prove Theorem B. As mentioned in the introduction, the goal is to find smooth structures $N$ and large subgroups $F<\operatorname{Out}(\pi)$ so that $\operatorname{Im}\Psi_{N}\cap F=1$. To this end, we consider the exotic smooth structures $N=M_{c,\phi}$ studied in [FJ93]. Here $M$ is hyperbolic, $c$ is a simple closed geodesic, and $\phi\in\operatorname{Diff}(S^{n-2})$. Choosing a framing $\iota:S^{1}\times D^{n-1}\rightarrow M$ of $c$, the manifold $M_{c,\phi}$ is defined as the quotient of

[TABLE]

by the identification $(x,v)\leftrightarrow\iota(x,\phi(v))$ for $(x,v)\in S^{1}\times S^{n-2}$.

We prove Theorem B in 3 steps.

4.1 Non-concordant smooth structures (Step 1)

Our mechanism for constructing $\alpha\in\operatorname{Out}(\pi)$ such that $\alpha\notin\operatorname{Im}\Psi_{N}$ is Theorem 8 below. Before we state it, recall some facts about smooth structures that will be used here and in the next subsection.

Smoothings of topological manifolds. By a smooth manifold $N$ we mean a topological manifold with a smooth atlas of charts $\mathbb{R}^{n}\supset U_{\alpha}\rightarrow N$ (which we call a smooth structure). If $N$ (resp. $M$) is a smooth (resp. topological) manifold and $h:N\rightarrow M$ is a homeomorphism, then we obtain a smooth structure on $M$ by pushforward. The map $h$ is called a marking. Two markings $h_{0}:N_{0}\rightarrow M$ and $h_{1}:N_{1}\rightarrow M$ determine the same smooth structure on $M$ if there is a diffeomorphism $g:N_{0}\rightarrow N_{1}$ so that $h_{1}g=h_{0}$.

Two smooth structures $N_{0},N_{1}$ on $M$ are concordant if there exists a smooth structure on $M\times[0,1]$ whose restriction to $M\times\{i\}$ is $N_{i}$ for $i=0,1$. The main fact about concordances that we use is that classifying concordance classes reduces to homotopy theory: there is a bijection between the set of concordance classes of smooth structures on $M$ and the set of based homotopy classes of maps $[M,\operatorname{Top}/O]$.

As remarked in [FJ93, §1], the concordance class of the smooth structure $M_{c,\phi}$ is independent of the choice of framing and is also independent of the choice of representative of the isotopy class $[\phi]\in\pi_{0}\operatorname{Diff}(S^{n-2})$.

Theorem 8** (non-concordant smooth structures).**

Let $M$ be a smooth closed manifold. Assume $M$ is stably parallelizable. Let $c_{1},\ldots,c_{\ell}$ be disjoint closed curves in $M$. Assume that there exists a homomorphism $\Delta:\pi_{1}(M)\rightarrow\mathbb{Z}^{\ell}$ such that $\Delta(c_{1}),\ldots,\Delta(c_{\ell})$ generate $\mathbb{Z}^{\ell}$. For any nontrivial isotopy class $[\phi]\in\pi_{0}\operatorname{Diff}(S^{n-2})$, no two of the smooth structures $M_{c_{1},\phi},\ldots,M_{c_{\ell},\phi}$ are concordant.

Proof.

Given a codimension-0 embedding $\lambda:X\rightarrow Y$ of open manifolds, we denote $\lambda^{\prime}$ the induced map of 1-point compactifications, obtained by collapsing $Y\setminus X$ to a point. Also $X_{+}$ denotes the space $X$ with a disjoint basepoint.

Let $\iota_{1},\ldots,\iota_{\ell}:S^{1}\times D^{n-1}\hookrightarrow M$ be framings of $c_{1},\ldots,c_{\ell}$. Use $\iota_{1},\ldots,\iota_{\ell}$ to define an embedding $\iota:\coprod_{\ell}S^{1}\times D^{n-1}\hookrightarrow M$. The induced collapse map has the form $\iota^{\prime}:M\rightarrow\bigvee_{\ell}\Sigma^{n-1}(S^{1}_{+})$. Consider the composition

[TABLE]

where the last map is induced from the obvious maps $\Sigma^{n-1}(S^{1}_{+})\simeq S^{n}\vee S^{n-1}\rightarrow S^{n-1}$. It suffices to show that the induced map

[TABLE]

is injective. This is because, under the bijection between concordance classes of smooth structures on $M$ and $[M,\operatorname{Top}/O]$, the concordance class of $M_{c_{j},\phi}$ corresponds to the map

[TABLE]

where $\pi_{j}$ collapses every sphere other than the $j$-th sphere to the basepoint, and $\hat{\phi}$ corresponds to $[\phi]\in\pi_{0}\operatorname{Diff}(S^{n-2})$ under the bijections $[S^{n-1},\operatorname{Top}/O]\simeq\Theta_{n-1}\simeq\pi_{0}\operatorname{Diff}(S^{n-2})$.

To show that $\hat{\iota}^{*}$ is injective, we use that $\operatorname{Top}/O$ is an infinite loop space. In particular, there exists a space $Y$ such that $\Omega^{n+\ell}Y\simeq\operatorname{Top}/O$, and for any space $A$, there are natural bijections $[A,\operatorname{Top}/O]\simeq[A,\Omega^{n+\ell}Y]\simeq[\Sigma^{n+\ell}A,Y]$. This allows us to view $\hat{\iota}^{*}$ as map

[TABLE]

This map can also be obtained by considering the embedding $\iota\times 1:\left(\coprod_{\ell}S^{1}\times D^{n-1}\right)\times D^{n+\ell}\hookrightarrow M\times D^{n+\ell}$ and the composition $\widehat{\iota\times 1}:\Sigma^{n+\ell}(M_{+})\xrightarrow{(\iota\times 1)^{\prime}}\bigvee_{\ell}\Sigma^{2n+\ell}(S^{1}_{+})\rightarrow\bigvee_{\ell}S^{2n+\ell-1}$, similar to before.

The homomorphism $\Delta$ is induced by a map $\delta:M\rightarrow T^{\ell}$ to the torus, and we can assume $\delta$ is smooth. Take a Whitney embedding $\epsilon:M\rightarrow D^{2n}$, and consider the induced embedding $\delta\times\epsilon:M\rightarrow T^{\ell}\times D^{2n}$. Since $M$ is a stably parallelizable, $M\subset T^{\ell}\times D^{2n}$ has trivial normal bundle $\nu_{M}\simeq\epsilon^{n+\ell}$. (To see this, observe that $TM\oplus\nu_{M}\simeq\epsilon^{2n+\ell}$. Since $M$ is stably parallelizable, $TM\oplus\epsilon\simeq\epsilon^{n+1}$, which implies that $\epsilon^{n+1}\oplus\nu_{M}\simeq\epsilon^{2n+\ell+1}$. Since $\text{rank}(\nu_{M})>\dim M$, this implies that $\nu_{M}$ is the trivial bundle by [KM63, Lem. 3.5].) Then there is an embedding $\kappa:M\times D^{n+\ell}\rightarrow T^{\ell}\times D^{2n}$.

Consider now the composition

[TABLE]

To prove the theorem, we show that the induced map

[TABLE]

is injective. First observe the homotopy equivalence $\Sigma^{2n}(T^{\ell}_{+})\sim\bigvee_{i=0}^{\ell}{\ell\choose i}S^{2n+i}$. This follows from general homotopy equivalences $\Sigma(A_{+})\sim\Sigma A\vee S^{1}$ and $\Sigma(A\times B)\sim\Sigma A\vee\Sigma B\vee\Sigma(A\wedge B)$. Since $\Delta(c_{1}),\ldots,\Delta(c_{\ell})$ generate $\pi_{1}(T^{\ell})$, the inclusion $\ell\>S^{2n+\ell-1}\subset\bigvee_{i=0}^{\ell}{\ell\choose i}S^{2n+i}$ is a right inverse to $p$, up to homotopy. This implies that $p^{*}$ is injective. ∎

4.2 Outer automorphisms not realized by diffeomorphisms (Step 2)

Next we apply Theorem 8 to give a criterion that guarantees that $\alpha\in\operatorname{Out}(\pi)$ is not in the image of $\Psi_{N}:\operatorname{Diff}(N)\rightarrow\operatorname{Out}(\pi)$.

Theorem 9** (obstruction to Nielsen realization).**

Let $M$ be a hyperbolic manifold and fix a simple closed geodesic $c$ in $M$. Let $N=M_{c,\phi}$ be an exotic smooth structure. Assume that $\alpha\in\operatorname{Isom}(M)\simeq\operatorname{Out}(\pi)$ is such that $M_{c,\phi}$ and $M_{\alpha(c),\phi}$ are not concordant. Then $\alpha\notin\operatorname{Im}\Psi_{N}$.

Proof.

Suppose for a contradiction that there is a diffeomorphism $f:N\rightarrow N$ such that $\Psi_{N}(f)=\alpha$.

Set $N_{0}=N$ and $N_{1}=M_{\alpha(c),\phi}$, and observe that $\alpha:M\rightarrow M$ induces a diffeomorphism $g_{1}:N_{0}\rightarrow N_{1}$. Define $g_{2}=g_{1}\circ f^{-1}$. Denoting $h_{i}:N_{i}\rightarrow M$ be the obvious homeomorphisms, the composition

[TABLE]

induces the identity on $\pi$ and is therefore homotopic to the identity. From this homotopy, we obtain a homotopy equivalence $H_{0}:M\times[0,1]\rightarrow M\times[0,1]$, which restricts to a homeomorphism on the boundary. By [FJ89b, Cor. 10.6], $H_{0}$ is homotopic rel boundary to a homeomorphism $H$. Then the composition

[TABLE]

defines a smooth structure on $M\times[0,1]$ whose restriction to $M\times\{i\}$ is $N_{i}$ for $i=0,1$, i.e. $N_{0}$ and $N_{1}$ are concordant. This contradicts our assumption, so $\alpha\notin\operatorname{Im}\Psi_{N}$. ∎

4.3 Examples (Step 3)

To complete the proof of Theorem B, we explain how to obtain examples of stably parallelizable $M$ that satisfy the assumptions of Theorems 8 and 9. This is the content of the following proposition.

Proposition 10**.**

Fix $n\geq 2$. For any $d\geq 2$, there exists a stably parallelizable hyperbolic manifold $M^{n}$, a geodesic $c$, a subgroup $F<\operatorname{Isom}(M)$ isomorphic to $\mathbb{Z}/d\mathbb{Z}=\langle\alpha\rangle$, and $\rho\in H^{1}(M)\simeq\operatorname{Hom}(H_{1}(M),\mathbb{Z})$ such that

[TABLE]

Consequently, the homomorphism $\Delta:H_{1}(M)\rightarrow\mathbb{Z}^{d}$ whose $i$-th coordinate is $\rho\circ\alpha^{-i}$ has the property that $\Delta(c),\ldots,\Delta(\alpha^{d-1}c)$ generate $\mathbb{Z}^{d}$.

In [Lub96], Lubotzky gave examples of hyperbolic $M$ (both arithmetic and non-arithmetic) with a surjection $\pi_{1}(M)\twoheadrightarrow F_{r}$ to a free group of rank $r\geq 2$. By passing to a cover, we can assume that $M$ is stably parallelizable [Sul79, pg. 553]. Proposition 10 is proved by passing to a further cover, using the general procedure of the following lemma.

Lemma 11**.**

Let $X$ be a CW-complex, and let $F_{r}$ denote a free group of rank $r\geq 2$. Assume there is a surjection $\pi_{1}(X)\twoheadrightarrow F_{r}$. Then for any $d\geq 2$, there exists a regular cover $Y\rightarrow X$ with deck group $\mathbb{Z}/d\mathbb{Z}=\langle\alpha\rangle$ and $c\in\pi_{1}(Y)$ and $\rho\in H^{1}(Y)$ satisfying (5).

Proof.

Take $F_{r}$ with generators $a_{1},\ldots,a_{r}$. Consider $F_{r}\twoheadrightarrow\mathbb{Z}/d\mathbb{Z}$ defined by $a_{1}\mapsto 1$ and $a_{i}\mapsto 0$ for $2\leq i\leq r$. Then $\ker[F_{r}\twoheadrightarrow\mathbb{Z}/d\mathbb{Z}]\simeq F_{k}$ with $k=1+d(r-1)$. It’s easy to compute $H_{1}(F_{k})$ as a $F=\mathbb{Z}/d\mathbb{Z}$-module:

[TABLE]

(For example, realize $1\rightarrow F_{k}\rightarrow F_{r}\rightarrow\mathbb{Z}/d\mathbb{Z}\rightarrow 0$ as a $(\mathbb{Z}/d\mathbb{Z})$-covering of graphs.) Then also $H^{1}(F_{k})\simeq\mathbb{Z}\{\beta_{1}\}\oplus\mathbb{Z}F\{\beta_{2},\ldots,\beta_{k}\}$, where $\beta_{i}$ is dual to $b_{i}$.

Let $Y\rightarrow X$ be the cover such that \pi_{1}(Y)=\ker\big{[}\pi_{1}(X)\twoheadrightarrow F_{r}\twoheadrightarrow\mathbb{Z}/d\mathbb{Z}\big{]}. Then $\pi_{1}(Y)\twoheadrightarrow F_{k}$, and $H_{1}(Y)\rightarrow H_{1}(F_{k})$ is $(\mathbb{Z}/d\mathbb{Z})$-equivariant. Choose $c\in\pi_{1}(Y)$ so that $c\mapsto b_{2}$ under $\pi_{1}(Y)\twoheadrightarrow F_{k}$, and define $\rho:\pi_{1}(Y)\twoheadrightarrow F_{k}\xrightarrow{\beta_{2}}\mathbb{Z}$. It’s easy to verify that $\rho$ satisfies (5). This proves the lemma. ∎