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.

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 {\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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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