We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts “like an automorphism” on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological prope...

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...

We derive presentations of the interval groups related to all quasi-Coxeter elements in Coxeter group type D n . Type is only infinite family finite that admits proper elements. The we obtain are over a set generators bijection with what call Carter generating set, and relations those defined by diagram together twisted cycle or commutator relator, depending on whether element not. proof based ...

Following Malcev [5], we will call a subgroup M of a group G finitely separable if for any element g ∈ G, not belonging to M, there exists a homomorphism φ of group G onto some finite group X such that gφ / ∈Mφ. This is equivalent to the statement that for any element g ∈ G \M, there exists a normal subgroup N of finite index in G such that g / ∈MN . A group G is subgroup separable if each of i...

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 connective constant of a graph is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. Strict inequalities are proved for connective constants of vertex-transitive graphs. First, the connective constant decreases strictly when the graph is replaced by a nontrivial quotient graph. Second, the connective constant increases strictly when a quasitransitiv...

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 ...