Positively Expansive Maps and Growth of Fundamental Groups

Abstract

ABSTRACT. In this note we will show that a positively expansive map of an arbitrary closed topological manifold is topologically conjugate to an expanding infra-nil-endomorphism.Let r be a group generated by {71,..., ik}-Then each 7 e T is represented as a word 7f117f2 ■ • -7^' and the number |pi| + IP2I + • • • + |p¡| is called the length of the word. The norm \\i\\ is defined as the minimal length of the word representing 7. As properties of the norm we know that ||7|| = ||7_1|| and ||77'|| < ||7|| + \\l'\\-If ¿1,..., 6k is another system of generators in T, then the corresponding norm || ||' is not necessarily equal to || ||. But there exists c > 0 such that for each 7 G T Let r be a finitely generated group and fix its generators. We denote by B(t), í > 0, the ball of radius t centered at the identity element e, and by #B(r) the number of elements in B(t).We say that T has polynomial growth if there are two positive numbers d and c such that for all balls B(t), t > 1, #5(i) < ctd. If T has a nilpotent subgroup of finite index then T has polynomial growth (Wolf [10]).The following is one interesting result for our investigation.THEOREM A (M. GroMOV [5]). // a finitely generated group T has polynomial growth, then T contains a nilpotent subgroup of finite index.Let L be a simply connected nilpotent Lie group with a left invariant Riemannian metric and denote by Aff (L) the group of transformations of L generated by the left translations and by all automorphisms from L onto itself. Let T C Aff(L) be a group which acts freely and discretely on L. If the quotient X = L/T is compact, then it is called an infra-nil-manifold. Each expanding automorphism L -» L which respects V induces an expanding map X -* X. Such maps are called an expanding infra-nil-endomorphism.From the following results together with Theorem A, M. Gromov [5] proved that an expanding differentiable map of an arbitrary closed C°° manifold is topologically conjugate to an infra-nil-endomorphism.

Observing expansive maps

“…Proof From we know that is conjugate to an expanding endomorphism. Then, the result follows by Theorem .…”

“…The main examples of positively expansive maps are Ruelle expanding maps [, § 7.26] and the one‐sided shift map on the Cantor set. It is remarkable that on compact manifolds every positively expansive map is conjugate to an infra‐nilmanifold expanding endomorphism (see ).…”

“…For example, they cannot be injective (see also ) and if a compact metric space admits a positively expansive map then the space has finite topological dimension (recently, in this result was generalized to actions of and mean dimension). Also, if the space is a compact manifold then every positively expansive map is conjugate to an infra‐nilmanifold expanding endomorphism, see . In particular, positively expansive maps on tori are conjugate to linear expanding endomorphisms.…”

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

“…Gromov [11] proved that an expanding differentiable map of a closed smooth manifold is topologically conjugate to an expanding infra-nil-endomorphism. Hiraide [14] proved that a positively expansive map of a closed topological manifold is topologically conjugate to an expanding infra-nil-endomorphism.…”

What an infra-nilmanifold endomorphism really should be

“…The discussion above shows that this is not a restriction. In ( [9]) and ( [11]), an infra-nilmanifold is defined as a quotient Γ\N , where Γ is a subgroup of the whole affine group Aff(N ) acting freely and properly discontinuously on N . This is not a correct definition, for in this case, the linear parts do not have to form a finite group and hence Γ need not be a virtually nilpotent group.…”