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Endomorphic presentations of branch groups. (English) Zbl 1044.20015

In the literature there are many examples of groups the presentations by generators and defining relations of which use endomorphisms and iterations under a set of substitutions on the generating set. Generalizing this concept the current paper introduces the notion of ‘\(L\)-presentations’. The latter is denoted by \(L=\langle S\mid Q|\Phi|R\rangle\), where \(S\) is an alphabet, \(Q\) and \(R\) are sets of reduced words in the free group \(F_S\), and where \(\Phi\) is a set of free group homomorphisms \(\phi\colon F_S\to F_S\). The group \(G_L\) defined by such an \(L\)-presentation is the factor group \(G_L=F_S/N\), where \(N\) is the normal closure in \(F_S\) of the subgroup \(\langle Q\cup\bigcup_{\phi\in\Phi^*}\phi(R)\rangle\), and where \(\Phi^*\) is the monoid generated by \(\Phi\). This \(L\)-presentation is ‘finite’ if \(S,Q,\Phi,R\) are finite. It is ‘ascending’ if \(Q\) is empty. It is ‘injective’ if, furthermore, the homomorphisms \(\phi\in\Phi\) are injective.
The paper studies the properties of such presentations and, in particular, proves Theorem 1.1. Let \(G\) be a finitely generated, contracting, semi-fractal, regular branch group. Then \(G\) is finitely \(L\)-presented. However, \(G\) is not finitely presented. The Schur multiplier of \(G\) has the form \(A\oplus B^\infty\) for finite-rank groups \(A,B\).

MSC:

20F05 Generators, relations, and presentations of groups
20E08 Groups acting on trees
20F65 Geometric group theory

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References:

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