Geometric group theory is a fairly new and active branch of mathematics that appeared at the end of last century, growing out of combinatorial group theory. The original focus of geometric group theory is the study of groups viewed as geometric objects, and nowadays it closely interacts with low-dimensional topology, differential geometry, ring theory, mathematical logic, dynamical systems, probability and K-theory, to mention a few.
La teoría geométrica de grupos es una rama bastante nueva y activa de las matemáticas que nació a finales del siglo pasado y surgió de la teoría combinatoria de grupos. El enfoque original de la teoría geométrica de grupos es el estudio de grupos vistos como objetos geométricos, y hoy en día interactúa estrechamente con la topología geométrica, la geometría diferencial, la teoría de anillos, la lógica matemática, los sistemas dinámicos, la probabilidad y la K-teoría entre otras.
20F65
(primary)
05C25; 20F67; 20E26
(secondary)
2.A (0.1)
2.B (0.1)
2.C (0.1)
A finitely generated subgroup $H$ of a finitely generated group $G$ is a virtual fiber subgroup if $G$ admits a finite index subgroup which surjects onto the integers and the kernel has finite index in $H$. This condition is very strong; it implies a number of nice properties of the subgroup, such as separability, but also imposes a number of geometric properties on the quotient $H\backslash G$. In this talk, I will discuss the extent to which these geometric properties characterise virtual fiber subgroups.
Right-angled Artin groups (RAAGs) play a central role in geometric group theory. In this talk, we introduce twisted right-angled Artin groups (T-RAAGs), a generalization of RAAGs. T-RAAGs are defined using a mixed graph: undirected edges [$a$ — $b$] impose the relation $ab = ba$, while directed edges [$a \rightarrow b$] give the Klein relation $aba = b$. We present a normal form for elements of T-RAAGs and utilize it to explore the geometric and algebraic similarities and differences between T-RAAGs and RAAGs.
Stallings foldings were introduced in 1983 as a tool to understand f.g. subgroups of free groups through finite objects called Stallings automata. Since then several generalisations of the automata have been developed for e.g. RAAGs or automatic groups. In 2017 Kharlampovich, Miasnikov and Weil described a way to build Stallings automata for certain subgroups of automatic groups, we now present some work in progress that adapt these ideas to understand semigroups inside automatic groups.
Joint work with John Britnell, Andrew Duncan, Dominik Francoeur, and Sarah Rees.
Let $F$ be the free group of finite rank $n$ and let $\Phi$ be an automorphism of $F$. A folkloric conjecture of Scott stated that the subgroup of elements of $F$ fixed by $\Phi$ has rank at most $n$. This was settled by Bestvina and Handel in 1992, for which they developed the analogous theory of Thurston's train-track maps in this context. We will discuss a new proof of Scott's conjecture based on $L^2$-homology.
Artin groups are defined from a set of generators $S$ and relations $aba\ldots= bab\ldots$, where the words are of the same length. Objects of great interest within the study of these groups are the standard parabolic subgroups. In this talk, we will present a problem related to the conjugation of these subgroups known as the ribbon conjecture. It asks whether two parabolic subgroups $P, Q$, we have that $gPg^{-1}=Q$ if and only if $g$ is the product of an element of $P$ and some special elements called ribbons.
The study of the rationality of $L^2$-Betti numbers has led to a rich theory in $L^2$-homology with deep implications in structural properties of groups. For decades it has been unclear if the strong Atiyah conjecture passes to free products. We will confirm that the strong and algebraic Atiyah conjectures are closed under the graph of groups construction provided that the edge groups are finite and show that the $\ast$-regular closure is a universal localization of the associated graph of rings
The compressed word problem is a variant of the classical word problem in which the input word is given as a context-free grammar that produces just one word. In this talk, we will introduce the basic notions of compression that are necessary to understand this problem and we will show the connection between the compressed word problem for a group and the classical word problem for its group of automorphisms. We will also discuss the strategy to solve the compressed word problem in an example.
The family of right-angled Artin groups (RAAGs) interpolates between free groups and free abelian groups. A group is said to algebraically fiber if it surjects onto $\mathbb{Z}$ with finitely generated kernel. This property is coarsely connected with the $L^2$-Betti numbers of the group, a powerful homological invariant.
In this talk we will present some partial results on the computation of the $L^2$-Betti numbers and the fibration properties of the (outer) automorphism group of a RAAG.
In this talk, we will generalize to certain Artin groups some results previously known for right-angled Artin groups. Firstly, we will show that the derived subgroup of an Artin group is free if and only if the group is coherent. Secondly, we will discuss finitely generated normal subgroups of coherent Artin groups, by showing that they are (mostly) co-(virtually abelian). Finally, we will talk about acylindrical hyperbolicity of their subgroups.
Joint work with Conchita Martínez Pérez.
We prove that torsion subgroups of groups defined by, C(6), C(4)-T(4) or C(3)-T(6) small cancellation presentations are finite. This follows from more general results about locally elliptic action on small cancellation complexes.
The McCullough-Miller space is a contractible simplicial complex that admits an action of the pure symmetric automorphisms of the free group, with stabilizers that are free abelian. It has been used to derive several cohomological properties of these groups, such as computing their cohomology ring and proving that they are duality groups. We will generalize the construction of McCullough-Miller to PSA of right-angled Artin groups, and use it to obtain some cohomological results about them.
Joint work with Conchita Martinez Perez and Richard Wade.
The close connections between dynamical properties of automorphisms of finitely generated free groups and geometric and algebraic aspects of (finitely generated free)-by-cyclic groups has led to several decades of fruitful research. In this talk I will briefly summarise some of what is known about (finitely generated free)-by-cyclic groups, I will discuss generalisations to the much larger family of free-by-cyclic groups and present some applications to the theory of one-relator groups.