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
651
questions
1
vote
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47 views
$V$-like actions of $V$
This continues my question about prefix-continuous bijections (since the answer was "yes").
Notation and conventions: Let $A$ be a finite alphabet and $L \subset A^*$ a language. Let $G$ be a group. ...
6
votes
1answer
128 views
Is there a prefix-continuous bijection between finite words and eventually zero words?
Let
$$ X = \{x \in \{0,1\}^{\omega} \;|\; \exists m: \forall i \geq m: x_i = 0\} $$
(one-way infinite eventually zero words). Let $\{0,1\}^*$ denote the finite (not necessarily nonempty) words over $\{...
9
votes
1answer
179 views
hyperbolic quotient of hyperbolic group
I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...
0
votes
0answers
68 views
Large gaps in the norm of a subgroup and its centraliser
Take an infinite finitely generated group $G$ with an infinite subgroup $N$ which has an infinite centraliser $Z = Z_G(N)$.
Let $S$ be some [symmetric] generating set of $G$ and for $g \in G$,
...
4
votes
0answers
157 views
Image of the mapping class group of surfaces into automorphism group?
Let $S_{g,p}^n$ be a compact oriented surface of genus $g$ with $p$ punctures and $n$ boundary components, and $\operatorname{Mod}(S)$ and $\operatorname{PMod}(S)$ be the mapping class group and the ...
0
votes
0answers
62 views
QI-closure of $\mathrm{NA}\times\mathrm{NA}$
Suppose we know the following about a class of groups $\mathcal{G}$.
If $G$, $H$ are f.g. r.p. nonamenable groups, then $G \times H \in \mathcal{G}$.
If $G \in \mathcal{G}$, $G$ is f.p., and $G$ is ...
8
votes
0answers
127 views
Reference request: Name or use of this group of diffeomorphisms of the disc
Let $k \in \{0,\infty\}$, $G\subseteq \operatorname{Diff}^k(D^n)$ be the set of diffeomorphisms $\phi:D^n\to D^n$ of the closed $n$-disc $D^n$ (with its boundary) satisfying the following:
$
\phi(S_r^...
3
votes
0answers
145 views
Convergence of Fuchsian groups and existence of suitable homeomorphisms
Let $(\Gamma_n)_n$ ($\subset PSL(2,\mathbb{R})$) be a sequence of discrete groups, if we say that $(\Gamma_n)_n$ converges to a group $\Gamma$ this means that there exist isomorphisms $\tau_n:\Gamma\...
8
votes
0answers
118 views
Membership problem in general linear group
This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am.
Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...
2
votes
0answers
65 views
Finitely generated uniformly amenable groups
Keller in "Amenable groups and varieties of groups" introduces uniformly amenable groups as groups such that
there is a function $a: ]0,1[ \times \mathbb{N} \to \mathbb{N}$ such that
for any finite ...
5
votes
0answers
123 views
For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?
Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$,...
12
votes
1answer
193 views
Powers of the Euler class, torsion free subgroup of Homeo($S^1$)
For any subgroup $G$ of $\text{Homeo}(S^1)$, we have the Euler class $\chi$ in the group cohomology $H^2(G;\mathbb{Z})$. One can think of this class as the pullback of the generator of $H^2(\mathrm{B}\...
15
votes
1answer
265 views
Can a torsion-free group be quasi-isometric to a torsion group?
I have looked around in the literature on group theory and geometric group theory and this looks to be an open question as far as I can tell (by torsion group, I mean as usual a group in which every ...
4
votes
1answer
89 views
Is a cosparse action on a CAT(0) cube complex an essential action?
Let $X$ be a CAT(0) cube complex.
(From Sageev and Wise's Cores for Quasiconvex actions)
A group $G$ acts cosparsely on a CAT(0)-cube complex $X$ if there exists a compact space $K$ and finitely many ...
3
votes
0answers
64 views
The divergence function and quasigeodesics in $\delta$-hyperbolic spaces
(From Bridson and Haefliger's Metric Spaces of Non-positive Curvature) Let $X$ be a metric space. A map $e: \mathbb{N} \rightarrow \mathbb{R}$ is said to be a divergence function for $X$ if the ...
4
votes
2answers
121 views
When existence of loxodromic, WPD elements implies an action is acylindrical
Definitions
Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries.
Recall that an action of $G$ on $X$ is called acylindrical if the ...
16
votes
1answer
252 views
Existence of a quasi-isometric residually finite group?
It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...
5
votes
1answer
147 views
The stabilizers of the canonical boundary action of hyperbolic groups
My question is that
Is every stabilizer of the canonical boundary action of a hyperbolic group on its Gromov boundary a finitely generated group?
I guess every stabilizer is a (finitely generated) ...
3
votes
1answer
93 views
Bounding the size of the conjugating elements given the Dehn function
I am learning a little bit about Dehn functions of group presentations and I came across a question that is probably pretty basic but that I was giving me trouble. I'll set some notation but ...
3
votes
0answers
50 views
A Jacobian of the Cayley graph
Let $\pi$ be a finitely generated discrete group. Let $F$ be a finitely generated free group with an epimorphism to $\pi$. Let $K$ be the kernel of the epimorphism. Consider the Abelianization of $K$. ...
3
votes
1answer
74 views
Quasi-isometries and E-unitary inverse semigroups
Let $S = \langle K\rangle$ be a finitely generated inverse semigroup, where $K \subset S$ is a fixed, finite and symmetric set of generators.
Preliminaries: Recall that we say that $s, t \in S$ are $\...
1
vote
0answers
36 views
Examples of groups admitting a proper $1$-cocyle for a bounded representation
A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...
4
votes
1answer
119 views
Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations
A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...
3
votes
2answers
261 views
On what proper Gromov hyperbolic space does a free product act?
Per Bowditch, a group is relatively hyperbolic if it acts geometrically finitely on a proper geodesic Gromov hyperbolic space.
A free product of two (or finitely many) finitely generated groups is ...
7
votes
1answer
221 views
Thickness and hierarchical hyperbolicity
Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here.
I've heard that it is open ...
1
vote
0answers
50 views
Quotient of Euclidean space with maximal volume growth
Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...
3
votes
0answers
147 views
Property (T) for pairs
I was reading the excellent book by Bekka, de la Harpe, and Valette (link at Bekka's page), and on its list of open questions I was looking at p300 Question 7.7:
$L\subset K\subset H\subset G$.$H,K,...
8
votes
2answers
348 views
Constant Martin kernel and amenability
Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...
4
votes
0answers
227 views
Commutator algorithm
Let $M \in \mathrm{SL}(2, \mathbb{Z}).$ Is there an efficient algorithm to write $M$ as a commutator (group commutator, not algebra commutator) [or fail if this is impossible]?
Addendum: answering ...
8
votes
0answers
165 views
Dualizing module for $\operatorname{Aut}(F_n)$
In The complex of free factors of a free group (pdf at Hatcher's page), Hatcher and Vogtmann defined a simplicial complex $FC_n$ called the ``complex of free factors'' of the free group $F_n$. They ...
3
votes
1answer
187 views
Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space
Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂𝑋 be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich https://...
9
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231 views
Finite quotients of surface braid groups
Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$.
There is a braid $A_{12} \in \Sigma_b$ ...
9
votes
0answers
210 views
Status of questions in “Group Actions on $\mathbb{R}$-trees”?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
2
votes
1answer
263 views
Is a product of Følner sets Følner?
Let $G$ be an amenable (countable, discrete) group and let $F_1,F_2,...,F_n,...$ and $G_1,G_2,...,G_n,...$ be two Følner sequences. Is the product sequence (i.e. the sequence $(H_n)$ where $H_n$ is ...
4
votes
1answer
171 views
Universal cover or Bass-Serre tree: difference between definitions given by Bass and Serre
Let $(\mathbb G,\Gamma)$ be a graph of groups.
A $G$-path from $u_0$ to $u$ is
$$g_0e_1g_1\cdots e_{n}g_n,$$
where $e_1\cdots e_{n}$ is a walk in $\Gamma$ from $u_0$ to $u$ and each $g_i\in G_{s(e_i)}$...
7
votes
2answers
200 views
Examples of non-cubulated hyperbolic groups
What is known regarding which hyperbolic groups are cubulated?
I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.
My impression is that not all ...
5
votes
0answers
78 views
Intuition behind the local limit theorem in hyperbolic groups
Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Denote by $X_n$ the induced random walk. Finally, let $p_n=\mu^{*n}(e)=P_e(X_n=e)$. The local limit ...
4
votes
1answer
229 views
Growth rates of surface groups
I'm looking for readable references on calculating the growth rates of surface groups.
There's some approach done briefly in page 159 of de la Harpe's "Topics in Geometric Group Theory", who cites:
...
11
votes
1answer
283 views
Is property FA of Serre known for $SL_2(\mathbb{Z}[i])$ and $SL_2(\mathbb{Z}[\zeta_3])$
In Serre's book Trees [Se, p. 68] it says:
3) For $SL_2$ the situation is different. It is clear that
$SL_2(\mathbf Z )$ does not have property (FA). It is the same with
$SL_2(A)$ when $A$ is ...
4
votes
0answers
73 views
Abelian-by-cyclic subgroups of exponential growth solvable groups
I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
3
votes
1answer
151 views
Asymmetry of outer space - injectivity radius
I'd like to ask a question on "Asymmetry of Outer Space" by Yael Algom-Kfir & Mladen Bestvina.
In Example $2$, page $4$ it says "Note that in this case the asymmetry can be explained by the fact ...
3
votes
1answer
336 views
Does Thompson's group F have Yu's property A?
As far as I know, it is unknown whether Thompson's group F has Yu's Property A (that is, whether it is exact) or not. See for instance this MO question. The question is said to be open at several ...
6
votes
1answer
203 views
Multiplication in Thompson's Group F
Does there exist an algorithmic way to multiply two elements of the Thompson Group F together? Specifically when looking at it from the perspective of pairs of binary trees. To multiply two elements ...
3
votes
1answer
416 views
virtual isomorphism of groups is an equivalence relation
I am reading the book: Geometric Group Theory by
Cornelia Druţu and Michael Kapovich With an Appendix by Bogdan Nica. https://www.math.ucdavis.edu/~kapovich/EPR/ggt.pdf. On page 125 we have:
*Def: We ...
5
votes
0answers
99 views
Examples of non-uniform lattices in products of trees
Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete.
So as a vague general question, what ...
8
votes
0answers
168 views
Subgroups of torsion-free hyperbolic groups versus subgroups of hyperbolic groups
Let $\mathcal{S}$ be the class of finitely presented torsion-free groups which occur as the subgroup of some hyperbolic group (so for every $G\in \mathcal{S}$ there exists a hyperbolic group $H$ such ...
2
votes
1answer
151 views
Equivalence of harmonic measures on hyperbolic groups
Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, ...
7
votes
2answers
246 views
What does the free action of a surface group on an R-tree look like?
Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, ...
4
votes
0answers
158 views
QI but not ME : not finitely presented groups!
I would like to know the examples of two groups which are not finitely presented and are quasi-isometric (QI), but they are not measured equivalent (ME) (in the sense of Gromov).
In the literature, ...
4
votes
1answer
341 views
Characterization of countable groups as groups with a left-invariant distance with finite balls
In their book "Metric geometry of locally compact groups", Yves Cornulier and Pierre de la Harpe gave, the following characterization of countable groups: $G$ is countable if and only if it has a ...
