Questions tagged [geometric-group-theory]
Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations
1
vote
0answers
16 views
Hilbert space compression of lamplighter over lamplighter groups
$C_2 \wr \mathbb{Z}$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space.
Question: Consider the group $C_2 \wr (C_2 \wr \mathbb{Z})$, what is ...
1
vote
0answers
105 views
Simple One-relator Groups [on hold]
One-relator groups, that is, groups which admit a finite presentation $\langle A \: | \: w=1 \rangle$ for some $w \in A^\ast$, are well studied objects in combinatorial group theory. Many abstract ...
5
votes
1answer
148 views
Which groups contain a comb?
The comb is the undirected simple graph with nodes
$\mathbb{N} \times \mathbb{N}$
where $\mathbb{N} \ni 0$ and edges
$$ \{\{(m,n), (m,n+1)\}, \{(m,0), (m+1,0)\} \;|\; m \in \mathbb{N}, n \in \mathbb{N}...
5
votes
0answers
99 views
Uniform versus non-uniform group stability
Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, ...
2
votes
1answer
78 views
Criterion for visuality of hyperbolic spaces
I am trying to understand the following sentence on p. 156 of Buyalo-Schroeder, Elements of asymptotic geometry: "Every cobounded, hyperbolic, proper, geodesic space is certainly visual."
Let $X$ be ...
9
votes
1answer
233 views
Is residual finiteness a quasi isometry invariant for f.g. groups?
A "residually finite group" is group for which the intersection of all finite index subgroups is trivial. Suppose $G$ and $G'$ are two quasi-isometric finitely generated groups. Does the residual ...
6
votes
0answers
107 views
Minimum Simple Burger-Mozes Type Group
Burger and Mozes constructed (Burger and Mozes - Lattices in products of trees) infinite, finitely presented, torsion-free simple groups which split as amalgams of two finitely generated free groups ...
3
votes
1answer
71 views
Non-collapsible complexes
I don't know much about simple homotopy theory, so maybe my question is quite trivial:
How does one prove that some complexes are contractible but non-collapsible?
there are few refinements of this ...
5
votes
0answers
153 views
Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups
A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
11
votes
1answer
217 views
Quasimorphisms and Bounded Cohomology: Quantitative Version?
Consider maps from a discrete group $\Gamma$ to the additive group $\mathbb{R}$. A function $f:\Gamma \to \mathbb{R}$ is called a quasimorphism if it is locally close to being a group homomorphism. ...
5
votes
1answer
186 views
Decidability: Presentations vs. Groups
This question is just a curiosity for me as a non-expert. Quite often we ask about decidability of various properties in a group. Often the answer is that the property is undecidable in general. ...
6
votes
0answers
92 views
Second bounded cohomology and normal subgroups
It may be a naive question, but:
If a finitely generated group has an infinite-dimensional second bounded cohomology group, does it imply that it contains "many" normal subgroups?
But "many", ...
7
votes
2answers
264 views
Bieberbach theorem for compact, flat Riemannian orbifolds
In his thesis, Bieberbach solved Hilbert 18 problem and
proved that any compact, flat Riemannian manifold is a
quotient of a torus. I need a reference to an orbifold version
of this result: any ...
8
votes
1answer
157 views
Second Bounded Cohomology of a Group: Interpretations
Suppose we have a group $\Gamma$ acting on an abelian group $V$. Then it is well-known that the second cohomology group $H^2(\Gamma,V)$ corresponds to equivalence classes of central extensions of $\...
1
vote
0answers
106 views
Word length norm in the symmetric group $\mathfrak{S}_r$
Consider on the symmetric group $\mathfrak{S}_r$ the generating system $\{\tau_i;\,1\le i\le r-1\}$ with $\tau_i = \langle i,i+1\rangle$ and the corresponding word length norm $N$. Now let $\tau\in\...
