Related papers: Algorithmically complex residually finite groups
We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as…
Our main result is to show that every infinite, countable, residually finite group $G$ admits a Hausdorff group topology which is neither discrete nor precompact.
We construct the first examples of residually finite amenable groups that are not Hilbert-Schmidt (HS) stable. We construct finitely generated, class 3 nilpotent by cyclic examples and solvable linear finitely presented examples. This also…
In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include…
In the paper the class of all solvable extensions of a filiform Leibniz algebra in the infinite-dimensional case is classified. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of residually solvable Leibniz algebra.…
Groups of finite type (also called finitely constrained groups), introduced by Grigorchuk, are known to be the closure of regular branch groups. This article explores many of their properties. Firstly, we prove that being finitely…
In [6] we proved that the universal theory of infinite free lattices is (algorithmically) decidable, leaving open the problem of decidability of the full theory of an (infinite) free lattice. We solve this problem by proving that, for every…
Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to…
We classify finite-dimensional complex Hopf algebras $A$ which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements $G(A)$ is abelian such that all prime divisors of the order…
We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\mathrm{FP}_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely…
We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with…
We establish the existence, finiteness, and uniqueness up to scaling of various isoperimetric profiles of a group, in all dimensions. We also show that these profiles all coincide in dimensions 4 and higher; in particular, the nth Dehn…
Ian Leary inquires whether a class of hyperbolic finitely presented groups are residually finite. We answer in the affirmative by giving a systematic version of a construction in his paper, which shows that the standard 2-complexes of these…
We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups…
The \emph{Entscheidungsproblem}, or the classical decision problem, asks whether a given formula of first-order logic is satisfiable. In this work, we consider an extension of this problem to regular first-order \emph{theories}, i.e.,…
A tubular group $G$ is a finite graph of groups with $\mathbb{Z}^2$ vertex groups and $\mathbb{Z}$ edge groups. We characterize residually finite tubular groups: $G$ is residually finite if and only if its edge groups are separable. Methods…
Stackability for finitely presented groups consists of a dynamical system that iteratively moves paths into a maximal tree in the Cayley graph. Combining with formal language theoretic restrictions yields auto- or algorithmic stackability,…
The first examples of formations which are arboreous (and therefore Hall) but not freely indexed (and therefore not locally extensible) are found. Likewise, the first examples of solvable formations which are freely indexed and arboreous…
Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…