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Question. Has a statement about groups ever been proved using the theory of semigroups?

By "a proof using the theory of semigroups" I do not mean that some steps in the proof are in fact statements about semigroups rather than groups. What I'm looking for are proofs which involve semigroups in a substantial way, say by constructing some semigroup related to the group of interest.

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At page 4 of J. Meakin's paper Groups and semigroups: connections and contrasts it is explained that

many properties of braid groups, Artin groups of finite type, Garside groups and the more general groups considered by Dehornoy are proved by a deep study of the associated monoid of positive elements. We refer to the papers of Dehornoy cited above for further references and details.

In the introduction of the same article it is also remarked that

central problems in finite semigroup theory (which is closely connected to automata theory and formal language theory) turn out to be equivalent or at least very closely related to problems about profinite groups

and moreover

the theory of inverse semigroups (i.e. semigroups of partial one-one functions) is closely tied to aspects of geometric and combinatorial group theory.

You can look at Meakin's paper and at the references contained therein for further details.

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The paper

Haar Measure and the Semigroup of Measures on a Compact Group, J. G. Wendel, Proceedings of the American Mathematical Society Vol. 5, No. 6 (Dec., 1954), pp. 923-929

proves the existence of Haar measure in compact groups using the existence of idempotents and minimal ideals in compact semigroups.

In more detail, the probability measures on a compact group form a compact semigroup with respect to convolution of measures and the weak* topology. One easily checks that the support of an idempotent measure is a closed subgroup and that the measure is a Haar measure on the support. If one takes an idempotent measure in the minimal ideal one checks the support is global and hence the measure is Haar measure for the group. Since all compact semigroups have a minimal ideal and the minimal ideal must have an idempotent, this yields Haar measure exists for compact groups.

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The statement that if $H_1,...,H_n$ are finitely generated subgroups of a free group, then $H_1...H_n$ is a closed subset in the profinite topology was proved by Ash in a semigroup reformulation of the result. Ribes and Zalesskii proved the theorem independently at around the same time using the theory of profinite groups acting on profinite trees. In fact, this fact was conjectured by two semigroup theorists/automata theorists motivated by a question in finite semigroup theory due to Rhodes.

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Karl Auinger and I initially proved using finite semigroups that if V is class of finite groups such that for every group $G$ in $V$, there is a non-trivial cyclic group of prime order $C$ such that the wreath product $C wr G$ belongs to V, then the product of finitely many finitely generated pro-V closed subgroup of a free group is pro-V closed. This includes an earlier result of Ribes and Zalesskii where V is assumed closed under extension but also includes classes like all finite groups of square free exponent.

We later came up with a semigroup free proof, but to some extent it is a veiled reinterpretation of the semigroup ideas in terms of graphs.

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It was proved by Birget, Margolis, Meakin and Weil that deciding whether a finitely generated subgroup $H$ of a free group is pure (meaning $H$ is closed under taking roots) is PSPACE-complete using semigroup theory.

The main idea is that to each Stallings graph of a finitely generated subgroup of a free group, you can associate a finite inverse semigroup given by generators. They proved that the subgroup is pure if and only if the inverse semigroup is aperiodic (meaning any subsemigroup which is a group is a trivial group). This can be checked in PSPACE. Then, they modified a classical construction in automata and finite semigroup theory showing that it is PSPACE-complete to decide if an automaton has an aperiodic transition semigroup, to prove completeness for purity.

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David Rees gave a beautiful proof that a cancellative monoid satisfying an Ore condition embeds in a group of fractions using inverse semigroup theory and the Green-Rees structure theory of D-classes. The key point is the Ore condition makes the inverse hull of this monoid E-unitary, which then gives the embedding into a group.

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There is a theorem saying that if we take a Zariski-dense subgroup $\Gamma$ in a semisimple real Lie group $G$, for example $G = \operatorname{SL}_d(\mathbb{R})$, the set of so-called "loxodromic" elements in $\Gamma$ is still Zariski-dense in $G$. (In the case of $G = \operatorname{SL}_d(\mathbb{R})$, "loxodromic" means "diagonalizable with eigenvalues of distinct modulus").

In order to prove it, one actually proves the same statement for semigroups. See Proposition 5.11 and Remark 5.14 in Y. Benoist and J.F. Quint's book "Random walks on reductive groups".

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