Articles of geometric group theory

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): $$v^{(SL(V)\cap Z)\gamma}:=\gamma(v)$$ this action is well defined, and if $W$ is a proper subspace of $V$ $$N_{PSL(V)}(W):=\{x\in PSL(V)\,:\, w^x\in W\quad\forall w\in W\}$$ Such normalizers […]

A Kleinian group has the same limit set as its normal subgroups'

It should be well known that a Kleinian group and all its normal (non-elementary) subgroups have the same limit set. Do you know any book/article where I could find the proof? Thank you.

The conjugacy problem of finitely generated free group

I would like references for algorithms solving the conjugacy problem in $F_n$ (the free group on $n$ generators)?

Free product as automorphism group of graph

Let $A$ and $B$ be two groups. We define following graph $X$. The set of vertices is the left cosets $gA$ and $gB$ where $g\in A*B$ (By $A*B$, I mean the free product of $A$ and $B$). The edges of the graph X correspond to the elements of $A*B$, and we use $e_g$ to denote […]

The only finite group which can act freely on even dimensional spheres is $C_2$.

I don’t know how to show this. Do I assume $G$ acts on $S^{2n}$ by homeomorphisms? Then, since $S^{2n}$ is Hausdorff I’d know $G$ acts freely and properly discontinuously, and since $\pi_1(S^{2n})={1}$ I’d have $\pi_1(X/G)\cong G$. But I’m not sure whether this is useful.

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 […]