Articles of geometric group theory

Connections of Geometric Group Theory with other areas of mathematics.

I’m a master’s student in the Turin University. At the end of my studies, I have to write a master thesis. My main interest is geometric group theory, but it is not a research area of the Turin’s mathematical department. My professors in algebra and geometry are principally interested in algebraic geometry, commutative algebra or […]

Isometries of a hyperbolic quadratic form

I am reading an article that says “The group of isometries (of a hyperbolic space) of a hyperbolic quadratic form in two variables is isomorphic to the semi-direct product $\mathbb{R} \rtimes \mathbb{Z}/2\mathbb{Z}$”. Could someone help me to understand that fact? First, it seems that the semi-direct product $\mathbb{R} \rtimes \mathbb{Z}/2\mathbb{Z}$ is equal to the direct […]

Name for Cayley graph of a semigroups?

I did Google search and can’t find a good answer. I thought I should ask experts here. Cayley graph is defined for groups. My question is: Is there a special name for the Cayley graph of semigroups?

Actions of Finite Groups on Trees

Any action of a finite group on a (non-empty) tree has a global fixed point (in the sense that there is a vertex fixed by all group elements or an edge fixed by all group elements). There is a hint which says we can consider the diameter of the corresponding orbit is minimal. However I […]

free subgroups of $SL(2,\mathbb{R})$

In the example section of the wikipedia article on the the Ping Pong lemma, you can see how to construct a free subgroup of $SL(2,\mathbb{R})$ with two generators $$ a_1 = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, \ \ \ \ \ a_2 = \begin{pmatrix} 1 & 0 \\ 2 & 1 […]

groups with infinitely many ends are not boundedly generated?

Recall that a group $G$ is boundedly generated if it can be written as a finite product of cyclic subgroups. And there are a lot of examples of groups that are (not) boundedly generated. I am wondering whether it is true that if $G$ has infinitely many ends, then it is not boundedly generated.

Are hyperbolic triangle groups hyperbolic?

This might be a silly question, but are hyperbolic triangle groups hyperbolic, in the sense of Gromov? By a hyperbolic triangle group, I mean a group given by a presentation, $$\langle a, b, c; a^p, b^q, c^r, abc\rangle$$ where $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$. I think they are, and it seems to be implied in some places, but nowhere […]

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn’t the case for the corresponding group with 4 generators, which is the famous Higman group. I know several direct algebraic proofs that $G$ is trivial. But they are all quite long and fiddly. I am not looking for […]

What is the easiest way to generate $\mathrm{GL}(n,\mathbb Z)$?

I’m searching for a way to generate the group $\mathrm{GL}(n,\mathbb Z)$. Does anyone have an idea? The intention of my question is that I am searching for an easy proof of the existence of the epimorphism: $\Phi:\mathrm{Aut}(F_n )\to \mathrm{Aut}(F_n/[F_n,F_n])=\mathrm{Aut}(\mathbb {Z}^n)=\mathrm{GL}(n,\mathbb {Z})$ I know that $\Phi$ is an canonical homomorphism since the commutator subgroup is characteristic […]

Integer lattice points on a sphere

Suppose we have a sphere centered at the origin of $\mathbb{R^{n}}$ with radius $r$. Are there known theorems that state the number of integer lattice points that lie on the sphere? It seems like this is something someone has studied so hopefully someone here could point me to some references. Also, consider the lattice points […]