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We study the following decision problem: given an exponential equation $a_1g_1^{x_1}a_2g_2^{x_2}\\dots a_ng_n^{x_n}=1$ over a recursively presented group G, decide if it has a solution with all $x_i$ in $\\mathbb {Z}$ . We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.

The work is devoted to the study of the combinatorial properties of determinism in a family of substitution complexes consisting of quadrilaterals glued together side-to-side. These properties are useful in the construction of algebraic structures with a finite number of defining relations. In particular, this method was used in constructing an infinite, finitely presented nilsemigroup satisfying the identity . This construction solves the problem posed by L.N. Shevrin and M.V. Sapir. This work investigates the possibility of coloring the entire family of complexes with a finite number of colors, for which the property of weak determinism holds: if the colors of three vertices of a given quadrilateral are known, then the color of the fourth vertex is uniquely determined, except in some cases of special arrangement of the quadrilateral. Even weak determinism is sufficient to construct a finitely presented nilsemigroup; when using this construction, the proof is shortened. Determinism properties help to correctly introduce defining relations in the semigroup of paths traversing the constructed complexes. The defining relations correspond to pairs of equivalent short paths. Properties of determinism have previously been studied in the context of tiling theory; in particular, Kari and Papasoglu constructed a set of square tiles that admits only aperiodic tilings of the plane and has the property of determinism: knowing the colors of two adjacent edges uniquely determines the colors of the remaining two edges.

Starting with a Grothendieck category G and a torsion pair t=(T,F) in G , we study the local finite presentability and local coherence of the heart H_t of the associated Happel–Reiten–Smalø t -structure in the derived category D(G) . We start by showing that, in this general setting, the torsion pair t is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those t for which H_t is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category G , which are general enough to include all locally coherent Grothendieck categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction, we characterize those torsion pairs t=(T,F) in a locally finitely presented G for which H_t is locally coherent in two cases: when the tilted t -structure in H_t is assumed to restrict to finitely presented objects, and when F is cogenerating. In the last part of the paper, we concentrate on the case when G is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations in this more classical setting, underlying connections with the notion of an elementary cogenerator.

We show that an N -complex of modules C is FP -injective if and only if C is N -exact and Z n i ( C ) is an FP -injective module for each i = 1 , 2 , ⋯ , N and each n ∈ Z by virtue of Gaussian binomial coefficients. Applications of this result go in three directions. Firstly, over a coherent ring, we prove that a bounded above N -complex C is FP -injective if and only if C is N -exact and C n is an FP -injective module for each n ∈ Z . Secondly, we obtain some examples of FP -injective N -complexes for some fixed integer N . Finally, we give a characterization of coherent rings.

For Higman’s sequence building operation and for any integer sequences set the subgroup is benign in a free group as soon as is benign in . Higman used this property as a key step to prove that a finitely generated group is embeddable into a finitely presented group if and only if it is recursively presented. We build the explicit analog of this fact, i.e., we explicitly give a finitely presented overgroup of and its finitely generated subgroup such that holds. Our construction can be used in explicit embeddings of finitely generated groups into finitely presented groups, which are theoretically possible by Higman’s theorem. To build our construction we suggest some auxiliary ‘‘nested’’ free constructions based on free products with amalgamation and HNN-extensions.

We consider certain properties of the semigroup S defined by the Presentation S = 〈a,b : a2ⁿ⁻¹ = 1,b2 = a2ⁿ⁻², ba = ab2ⁿ⁻¹-1〉, (n ≥ 3).

In this short note, I present a very quick review of the peculiarities of dimension four in geometric topology. I consider, in particular, the role of geometric simple connectivity (which means handle decomposition without handles of index one) for both closed manifolds and open manifolds and for finitely presented groups, together with some of recent developments in geometric group theory.