Antonio Machì

Amenable groups and cellular automata

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On the entropy of regular languages

Let L be an irreducible regular language. Let W be a non-empty set of words (or sub-words) of L and denote by LW = {v ∈ L:w ⊏ v, ∀w ∈ W} the language obtained from L by forbidding all the words w in W. Then the entropy decreases strictly: ent(LW) < ent(L). In this note we present a new proof of this fact, based on a method of Gromov, which avoids the Perron-Frobenius theory. This result applies to the regular languages of finitely generated free groups and an additional application is presented.

Il gruppo di Grigorchuk di crescita intermedia

An explicit description of the 2-group of intermediate growth found by Grigorchuk is given. We also give a few preliminary notions concerning finitely generated groups and their growth functions.

Generators and Relations

We recall that the subgroup 〈S〉 of a group G generated by a set S of elements of G is the set of all products

Solution to the exercises

a −1 b and ba −1 solve the two equations. Conversely, if ax = a has a solution, then there exists e r such that ae r = a; if b ∈ G, then b = ya, some y, and therefore be r = (ya) e r = y(ae) r = ya = b; hence e r is neutral on the right for all b ∈ G, and similarly there exists e l , neutral on the left. Multiplication of the two gives e l e r = e l , e l e r = e r , and ii) follows. As for iii), solving ax = e and ya = e, if ax = az we have y(ax) = y(az) from which (ya)x = (ya)z, i.e. ex = ez and x = z; hence the right inverse is unique, and so is the left inverse. If ax = e, then xa = exa = yaxa = y(ax)a = yea = ya = e and ax = e = xa.

Nilpotent Groups and Solvable Groups

There are two important properties of groups that are stronger than commutativity: they are solvability and nilpotence. Solvable1 groups are obtained by forming successive extensions of abelian groups; nilpotent groups lie midway between abelian and solvable groups.

The Euclidean algorithm, the Chinese remainder theorem and interpolation

Let m be an arbitrary integer number (positive, negative or zero), n a positive integer, and let

Group Actions and Permutation Groups

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