Laura Ciobanu

Solution Sets for Equations over Free Groups are EDT0L Languages

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to

Conjugacy growth series and languages in groups

\mathsf{NSPACE}(n\log n)

SOFIC GROUPS: GRAPH PRODUCTS AND GRAPHS OF GROUPS

here.

RAPID DECAY IS PRESERVED BY GRAPH PRODUCTS

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. 2010 Mathematics Subject Classification: 20F65, 20E45.

Formal Conjugacy Growth in Acylindrically Hyperbolic 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.

Classes of Groups Generalizing a Theorem of Benjamin Baumslag

We prove that the rapid decay property (RD) of groups is preserved by graph products defined on finite simplicial graphs.

Geodesic growth in right-angled and even Coxeter groups

Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivins conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivins conjecture for finitely generated acylindrically hyperbolic groups and prove a formal language version of it, namely that no set of minimal length conjugacy representatives can be unambiguous context-free.

On the asymptotics of visible elements and homogeneous equations in surface groups

Benjamin Baumslag proved that being fully residually free is equivalent to being residually free and commutative transitive (CT). Gaglione and Spellman and Remeslennikov 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 Tarski problems. In this article, we provide new examples of groups for which Benjamin Baumslags theorem is true, that is, we consider classes of groups 𝒳 for which a group is fully residually 𝒳 if and only if it is residually 𝒳 and commutative transitive. We show that this is true for many important classes of groups, including those of free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups (done by Kharlamapovich and Myasnikov), and one-relator groups with only odd torsion. Furthermore, we show that many of the properties discussed here are closed under taking free products. We then consider the classes of groups 𝒳 for which Baumslags conditions are also equivalent to being universally 𝒳.

The equation xp yq = zr and groups that act freely on L-trees

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.

POLYNOMIAL-TIME COMPLEXITY FOR INSTANCES OF THE ENDOMORPHISM PROBLEM IN FREE GROUPS

Let F be a group whose abelianization is Z k , k ≥ 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.