** Towers of regular self-covers and linear endomorphisms of tori **

**1. Self-covers **

Which closed manifolds admit self-covers?

**Standard examples**.

- Tori, with linear maps . Degree is .
- 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 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 admits an expanding map (with respect to some Riemannian metric), then is an infrnilmanifold.

** 1.1. Residual self-covers **

We suggest a topological analogue of expansion.

Say a cover is residual if iterates converge to the universal cover, i.e. every loop eventually unwinds.

**Question**. If has a residual finite self-cover, is 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 be a finitely generated group such that has a subgroup

Definition 2 (Nekrashevych-Pete)A finitely generated group is strongly scale invariant if is has a self-homomorphism with finite index image, and such that

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 , with trivial intersection, would suffice, but Nekrashevych-Pete showed this to be false.

One can ask more.

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

This is the case for the Baumslag-Solitar examples . The endomorphims are of the form multiplication by on the first factor.

**2. Result **

Definition 3A finite self-cover is strongly regular if all iterates are regular covers.

Theorem 4Let be a strongly regular self-cover. Then there exists a free abelian group and an epimorphism

such that induces an isomorphim on ker and a linear endomorphism of of determinant equal to the degree of .

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

Corollary 5If is residual and strongly regular, then is abelian.

Indeed, in this case, for all , so it must be trivial.

**3. Proof **

I explain the easier result where is merely torsion free nilpotent. Let be the profinite completion of . Then still has finite index image.

**Example**. . Multiplication by 2 is an example.

Theorem 6 (Reid 2014)Let be a profinite embedding of type . Let be an open embedding. Then is isomorphic to a semi-direct product , where 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 of nilpotent -adic groups and a group of bounded exponent.

**Example**. equipped with multiplication by .

** 3.1. Proof of theorem 1 **

We know that . First show the product is direct. Then that 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 is a self-cover has some geometric origin, is a fiber bundle with nilmanifold fibers ? For instance, is an endomorphism of some geometric structure, or is Anosov, or is holomorphic.

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