Hi, I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct …

EDIT: Mark Sapir pointed a reference (in the comments) giving a lower bound of $2^{1/4}$ for the minimal rate. Is this the state of art? The third question remains unanswered. If …

Dear friends, I am only a theoretical physicist. However, the answer to this question is relevant for emergence of space-time from a quantum cellular automaton (in the future I wi …

What is the role of the theory of Automatic Groups in the history of Geometric Group Theory? Motivation: When I read through the "Word Processing in Groups" I was amazed by the s …

Let $\Gamma$ be a group generated by symmetric finite set $S$ and acting on $X$. The Schreier graph of the action is the graph with vertex set $X$ and $(x,y)$ is an edge if there i …

Disclaimer : I suspect the question I am about to ask is really hard, but I just want to know the status of such questions. Thanks to Milnor, we know that the $\pi_1$ any compact …

Does anybody know the actions of Thompson group F which are not conjugate to the standard one? Motivation is to find actions such that the Schreier graph of the action does not co …

( I apologize for only indicating some easy to find references, but new users are not allowed to link more than five). This is very speculative, but: Question: Is there a reformul …

If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$?

Denote by $F_n$ the free group of rank $n$. We say that an automorphism $\phi\in Aut(F_n)$ is geometric if there exists a surface with boundary $M$ and a homeomorphism $h\colon M\t …

We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set. I have been co …

Does there exist a non trivial homomorphism from Thompson's group T to a linear group?

Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube comple …

Hi all, This is a question about the groups $H_{n,s}$ introduced by Völklein in his book "Groups as Galois groups", §7.1, and defined as follows: let $N$ be the intersection of al …

Consider a finitely generated group. Assume that the first Betti number of the ball of radius n in the Cayley graph is at most polynomial in n. This property is satisfied by free g …