Searcharxiv⌕ Search

arXiv subjects

Inna Bumagin

Publications and source records attributed to Inna Bumagin.

7 recordsLinked to original sources
arXiv2018

On fully residually-$\mathcal{R}$ groups

We consider the class $\mathcal{R}$ of finitely generated toral relatively hyperbolic groups. We show that groups from $\mathcal{R}$ are commutative transitive and generalize a theorem proved by Benjamin Baumslag to this class. We also discuss two definitions of (fully) residually-$\mathcal{C}$ groups and prove the equivalence of the two definitions for $\mathcal{C}=\mathcal{R}$. This is a generalization of the similar result obtained by Ol'shanskii for $\mathcal{C}$ being the class of torsion-free hyperbolic groups. Let $Γ\in\mathcal{R}$ be non-abelian and non-elementary. We prove that every finitely generated fully residually-$Γ$ group embeds into a group from $\mathcal{R}$. On the other hand, we give an example of a finitely generated torsion-free fully residually-$\mathcal{H}$ group that does not embed into a group from $\mathcal{R}$; $\mathcal{H}$ is the class of hyperbolic groups.

math.GR↗
arXiv2014

Effective coherence of groups discriminated by a locally quasi-convex hyperbolic group

We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $Γ$ is effectively coherent: that is, presentations for finitely generated subgroups can be computed from the subgroup generators. We study $G$ via its embedding into an iterated centralizer extension of $Γ$, and prove that this embedding can be computed. We also give algorithms to enumerate all finitely generated groups discriminated by $Γ$ and to decide whether a given group, with decidable word problem, is discriminated by $Γ$. If $Γ$ may have torsion, we prove that groups obtained from $Γ$ by iterated amalgamated products with virtually abelian groups, over elementary subgroups, are effectively coherent.

math.GR↗
arXiv2014

Time complexity of the conjugacy problem in relatively hyperbolic groups

If $u$ and $v$ are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok. Bridson and Haefliger showed that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in polynomial time in each parabolic subgroup. We also prove that if $u$ and $v$ are two conjugate hyperbolic elements of a relatively hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ is linear in terms of their lengths.

math.GR↗
arXiv2005

Isomorphism problem for finitely generated fully residually free groups

We prove that the isomorphism problem for finitely generated fully residually free groups is decidable. We also show that each finitely generated fully residually free group G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms Out(G).

math.GR↗
arXiv2004

The conjugacy problem for relatively hyperbolic groups

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this assertion. We conclude that the conjugacy problem is solvable for fundamental groups of complete, finite-volume, negatively curved manifolds, and for finitely generated fully residually free groups.

math.GR↗
arXiv2004

On definitions of relatively hyperbolic groups

The purpose of this note is to provide a short alternate proof that (combined with a theorem proven by Szczepanski) shows that a group which is relatively hyperbolic in the sense of the definition of Gromov is relatively hyperbolic in the sense of the definition of Farb.

math.GR↗