## Notes of Wouter Van Limbeek’s Cambridge lecture 08-05-2017

Towers of regular self-covers and linear endomorphisms of tori

1. Self-covers

Which closed manifolds ${M}$ admit self-covers?

Standard examples.

• Tori, with linear maps ${A}$. Degree is ${|det(A)|}$.
• Nilmanifolds, with automorphisms.
• certain fiber bundles with torus or nilmanifold fibers.

These are all examples in dimension 2 (Euler characteristic is multiplicative). Idem in dimension 3 (Tollefson, Mess, Yu-Wang 1999).

In dimension 4, there are other examples. Manifolds with fundamental group a Baumslag-Solitar ${BS(1,k)}$ have been classified by Hambleton-Kreck-Teichner, some of them double cover themselves.

Gromov’s polynomial growth theorem has the following consequence:

Theorem 1 (Gromov) If a closed manifold ${M}$ admits an expanding map (with respect to some Riemannian metric), then ${M}$ is an infrnilmanifold.

1.1. Residual self-covers

We suggest a topological analogue of expansion.

Say a cover ${f:M\rightarrow M}$ is residual if iterates ${f^k}$ converge to the universal cover, i.e. every loop eventually unwinds.

Question. If ${M}$ has a residual finite self-cover, is ${f}$ infranil ?

1.2. Strongly scale invariant groups

There is a purely group-theoretic analogue. Observe that a self-cover induces an map on fundamental groups whose image has finite index.

Question. Let ${\Gamma}$ be a finitely generated group such that ${\Gamma}$ has a subgroup

Definition 2 (Nekrashevych-Pete) A finitely generated group is strongly scale invariant if is has a self-homomorphism ${\phi:\Gamma\rightarrow \Gamma}$ with finite index image, and such that

$\displaystyle \begin{array}{rcl} \bigcap_k \phi^k(\Gamma)=\{1\}. \end{array}$

They ask wether strongly scalar invariant implies virtually nilpotent.

Previously Benjamini thought that existence of a chain of nested subgroups of finite index, all isomorphic to ${\Gamma}$, with trivial intersection, would suffice, but Nekrashevych-Pete showed this to be false.

Question. Is a self-cover somehow induced by an endomorphism of a nilpotent group?

This is the case for the Baumslag-Solitar examples ${\Gamma={\mathbb Z}[1/k]\times_k {\mathbb Z}}$. The endomorphims are of the form multiplication by ${d}$ on the first factor.

2. Result

Definition 3 A finite self-cover is strongly regular if all iterates ${f^k}$ are regular covers.

Theorem 4 Let ${f:M\rightarrow M}$ be a strongly regular self-cover. Then there exists a free abelian group ${A}$ and an epimorphism

$\displaystyle \begin{array}{rcl} 1\rightarrow ker(q)\rightarrow \Gamma\rightarrow A\rightarrow 1 \end{array}$

such that ${f}$ induces an isomorphim on ker${(q)}$ and a linear endomorphism of ${A}$ of determinant equal to the degree of ${f}$.

Question. Assume that ${M}$ is nonpositively curved and admits a self-cover. Then is ${M}$ covered by a product with a torus factor?

Corollary 5 If ${f:M\rightarrow M}$ is residual and strongly regular, then ${\pi_1(M)}$ is abelian.

Indeed, in this case, ${ker(q)\subset \phi^k(\Gamma)}$ for all ${k}$, so it must be trivial.

3. Proof

I explain the easier result where ${A}$ is merely torsion free nilpotent. Let ${\hat \Gamma}$ be the profinite completion of ${\Gamma}$. Then ${\hat\phi:\hat\Gamma\rightarrow\hat\Gamma}$ still has finite index image.

Example. ${\hat{\mathbb Z}=\prod_p {\mathbb Z}_p}$. Multiplication by 2 is an example.

Theorem 6 (Reid 2014) Let ${G}$ be a profinite embedding of type ${(F)}$. Let ${\phi:G\rightarrow G}$ be an open embedding. Then ${G}$ is isomorphic to a semi-direct product ${C\times Q}$, where ${C}$ is a contraction group, i.e. it admits an automorphism whose iterates bring everything asymptotically to the origin.

Theorem 7 (Gloeckner-Willis 2006) Any contraction group is a product of a finite product ${N\times F}$ of nilpotent ${p}$-adic groups and a group of bounded exponent.

Example. ${F=F_p[[t]]}$ equipped with multiplication by ${t}$.

3.1. Proof of theorem 1

We know that ${\hat \Gamma\simeq (N\times F)\times Q}$. First show the product is direct. Then that ${F}$ is trivial. This used the abelianization and the description of abelian groups of finite exponents (Prufer).

4. Kahler manifolds and holomorphic maps

Here is a vague question. If ${f:M\rightarrow M}$ is a self-cover has some geometric origin, is ${M}$ a fiber bundle with nilmanifold fibers ? For instance, ${f}$ is an endomorphism of some geometric structure, or ${f}$ is Anosov, or ${f}$ is holomorphic.

Theorem 8 If ${M}$ is Kahler and ${f}$ is strongly regular holomorphic self-cover, then ${M}$ is finitely covered by a product with a complex torus factor and a Kahler factor.