Related papers: The membership problem for 3-manifold groups is so…

We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a…

The following theorem is proved: Let $G$ be a finite group and $\pi_e(G)$ be the set of element orders in $G$. If $\pi_e(G) \cap \{2\}=\emptyset$; or $\pi_e(G) \cap \{3, 4\}=\emptyset$; or $\pi_e(G) \cap \{3,5\}=\emptyset$, then $G$ is…

We use language theory to study the rational subset problem for groups and monoids. We show that the decidability of this problem is preserved under graph of groups constructions with finite edge groups. In particular, it passes through…

Every countable group $G$ can be embedded in a finitely generated group $G^*$ that is hopfian and complete, i.e. $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is…

We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble.

We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization as quasi-geodesic monoids, and show that their word problem is rational (as a…

We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem…

We prove that the profinite completion of the fundamental group of a compact 3-manifold $M$ satisfies a Tits alternative: if a closed subgroup $H$ does not contain a free pro-$p$ subgroup for any $p$, then $H$ is virtually soluble, and…

This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial…

One method for obtaining every closed orientable 3-manifold is as branched covering of the 3-sphere over a link. There is a classical topological result showing that the minimun possible number of sheets in the covering is three. In this…

Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its…

This note surveys recent progress toward the profinite rigidity of orientable finite-volume hyperbolic 3-manifolds. Beginning in a brief review of some basic settings of profinite completion and rigidity of general groups, we state the…

We prove that any finitely generated one ended group has linear end depth. Moreover, we give alternative proofs to theorems relating the growth of a finitely generated group to the number of its ends.

We study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. In the case that M is reducible we build a contractible space parametrising the systems of reducing spheres. We use this…

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal…

We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.

We show that for a representation of the fundamental group of a triangulated closed 3-manifold (not necessarily hyperbolic) into $\PSL$ so that any edge loop has non-trivial image under the representation, there exist uncountably many…

The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership problem, as well as the conjugacy problem, for…

We investigate the Membership Problem for hypergeometric sequences: given a hypergeometric sequence $\langle u_n \rangle_{n=0}^\infty$ of rational numbers and a target $t \in \mathbb{Q}$, decide whether $t$ occurs in the sequence. We show…

Given a finitely generated group $G$ that is relatively finitely presented with respect to a collection of peripheral subgroups, we prove that every infinite subgroup $H$ of $G$ that is bounded in the relative Cayley graph of $G$ is…