We link distinct concepts of geometric group theory and homotopy through underlying combinatorics. For a flag simplicial complex $K$, we specify necessary sufficient combinatorial condition for the commutator subgroup $RC_K'$ right-angled Coxeter group, viewed as fundamental real moment-angle $\mathcal{R}_K$, to be one-relator group; Pontryagin algebra $H_*(\Omega \mathcal{Z}_K)$ algebra. also ...

Garside groups are a natural lattice-theoretic generalisation of the braid and spherical type Artin–Tits groups. Here we show that class is closed under some free products with cyclic amalgamated subgroups. We deduce every tree product infinite group. Moreover, study those HNN extensions as well. Using theorem Pietrowski, conclude this paper by stating non-cyclic one-relator group if only its c...

We develop a version of Bass-Serre theory for Lie algebras (over field k) via homological approach. define the notion fundamental algebra graph and show that this construction yields Mayer-Vietoris sequences. extend some well known results in group to N-graded algebras: example, we one relator are iterated HNN extensions with free bases which can be used cohomology computations apply sequence g...

The substitution acts on words (finite sequences) over this alphabet as well as on infinite sequences. It is easy to observe that τ has a unique invariant sequence ω that is the limit of words τ (a), k = 1, 2, . . . . Let Ω be the smallest closed set of one-sided infinite sequences over the alphabet {a, b, c, d} that contains ω and is invariant under the shift σ (σ acts on sequences by deleting...

Note that the existence of free subgroups in G̃ for n > 3 follows immediately from the free subgroup theorem for one-relator groups. Thus, Theorem 1 is nontrivial only for n = 2. The most difficult case is n = 1. An important role in this situation is played by the exponent sum of the generator in the relator. A word w = ∏ git εi ∈ G ∗ 〈t〉∞ is called unimodular if ∑ εi = 1. If the exponent sum o...

In this note we present a proof of a theorem to be found in Ehrig and Mahr [3). The theorem states that a relation constructed from a given function is a congruence relation if that function is a homomorphism: we go on to generalise this result to the relational homomorphisms treated by the first author in [1]. Specifically, we prove that. a congruence relation can be constructed from a relatio...