Sofic groups: graph products and graphs of groups
We prove that graph products of sofic groups are sofic, as are graphs of groups for which vertex groups are sofic and edge groups are amenable.
math.GR↗
arXiv subjects
Laura Ciobanu
Publications and source records attributed to Laura Ciobanu.
Classes of Groups Generalizing a Theorem of Benjamin Baumslag
In [BB] Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman [GS] and Remeslennikov [Re] showed that this is also equivalent to being universally free, that is, having the same universal theory as the class of nonabelian free groups. This result is one of the cornerstones of the proof of the Tarksi problems. In this paper we extend the class of groups for which Benjamin Baumslag's theorem is true, that is we consider classes of groups $\X$ for which being fully residually $\X$ is equivalent to being residually $\X$ and commutative transitive. We show that the classes of groups for which this is true is quite extensive and includes free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups (done in [KhM]), and one-relator groups with only odd torsion. Further, the class of groups having this property is closed under certain amalgam constructions, including free products and free products with malnormal amalgamated subgroups. We also consider extensions of these classes to classes where the equivalence with universally $\X$ groups is maintained.
math.GR↗
The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups
The general {\bf surface group conjecture} asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in [FKMRR]. First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if $G$ is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then $G$ is free, a surface group or a solvable Baumslag-Solitar Group. Further combining results in [FKMRR] on Property IF with a theorem of H. Wilton [W] and results of Stallings [St] and Kharlampovich and Myasnikov [KhM4] we show that Surface Group Conjecture C proposed in [FKMRR] is true, namely: If $G$ is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then $G$ is a surface group.
math.GR↗
Geodesic growth in right-angled and even Coxeter groups
The objective of this paper is to detect which combinatorial properties of a regular graph can completely determine the geodesic growth of the right-angled Coxeter or Artin group this graph defines, and to provide the first examples of right-angled and even Coxeter groups with the same geodesic growth series.
math.GR↗
Conjugacy growth series and languages in groups
In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.
math.GR↗
Rapid Decay is Preserved by Graph Products
We prove that the rapid decay property (RD) of groups is preserved by graph products defined on finite simplicial graphs.
math.GR↗
On the asymptotics of visible elements and homogeneous equations in surface groups
Let $F$ be a group whose abelianization is $\Z^k$, $k\geq 2.$ An element of $F$ is called visible if its image in the abelianization is visible, that is, the greatest common divisor of its coordinates is 1. In this paper we compute three types of densities, annular, even and odd spherical, of visible elements in surface groups. We then use our results to show that the probability of a homogeneous equation in a surface group to have solutions is neither 0 nor 1, as the lengths of the right- and left-hand side of the equation go to infinity.
math.GR↗
The monomorphism problem in free groups
Let $F$ be a free group of finite rank. We say that the monomorphism problem in $F$ is decidable if for any two elements $u$ and $v$ in $F$, there is an algorithm that determines whether there exists a monomorphism of $F$ that sends $u$ to $v$. In this paper we show that the monomorphism problem is decidable and we provide an effective algorithm that solves the problem.
math.GR↗