Related papers: Finding non-trivial elements and splittings in gro…
We show that an infinite group is definable in any non trivial geometric $C$-minimal structure which is definably maximal and does not have any definable bijection between a bounded interval and an unbounded one in its canonical tree. No…
Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups. It turns out that Stallings'…
We establish {\em{virtual surjection to pairs}} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,...,\Gamma_n$ are finitely presented and $S<\Gamma_1\times...\times\Gamma_n$ projects…
The isomorphism problem for infinite finitely presented groups is probably the hardest among standard algorithmic problems in group theory. Classes of groups where it has been completely solved are nilpotent groups, hyperbolic groups, and…
Let $\mathcal{C}$ be a class of finite groups closed for subgroups, quotients groups and extensions. Let $\Gamma$ be a finite simplicial graph and $G = G_{\Gamma}$ be the corresponding pro-$\mathcal C$ RAAG. We show that if $N$ is a…
We construct examples of finitely generated groups L that have non-trivial actions on $\mathbb{R}$-trees but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto L does have…
We show the existence of finitely presented torsion-free groups with decidable word problem that cannot be embedded in any finitely generated group with decidable conjugacy problem. This answers a well-known question of Collins from the…
A derived version of Maschke's theorem for finite groups is proved: the derived categories, bounded or unbounded, of all blocks of the group algebra of a finite group are simple, in the sense that they admit no nontrivial recollements. This…
We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated…
Moufang loops are one of the best-known generalizations of groups. There is only one countable family of nonassociative finite simple Moufang loops, arising from the split octonion algebras. We prove that every member of this family is…
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 $G$…
Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation…
Every nonabelian finite simple group of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, has a presentation with a bounded number of generators and relations and total length $O(\log n…
We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be…
The word problem of a group is a very important question. The word problem in the braid group is of particular interest for topologists, algebraists and geometers. In previouse article we have looked at the braid group from a topological…
We show that every finitely generated cohomologically trivial module over $RG$, where $G$ is a finite $p$-group and $R$ is a $p$-adic ring, splits as the direct sum of a finite cohomologically trivial $RG$-module and a free $RG$-module.…
We show how to construct all the extensions of left braces by ideals with trivial structure. This is useful to find new examples of left braces. But, to do so, we must know the basic blocks for extensions: the left braces with no ideals…
In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary…
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability.…
Suppose that G is a nontrivial torsion-free group and w is a word in the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from w by erasing all letters belonging to G is not a proper power in the free group…