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
987 questions
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3
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Subgroups of RAAGs vs. subgroups of RACGs
Is a (finitely generated) torsion-free subgroup of a right-angled Coxeter group isomorphic to a subgroup of a right-angled Artin group?
It is well-known from the theory of special cube complexes that ...
- 8,401
10
votes
2
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444
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Sequence of epimorphisms of residually finite groups stabilizes
Let $G_1 \to G_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize? That is, are all but finitely many epimorphisms actually ...
- 435
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2
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853
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Examples of hyperbolic groups with non-hyperbolic subgroups
In a previous question, I asked about hyperbolic groups in which every finitely generated subgroup is hyperbolic. I am now curious about the reverse question: what are some examples of hyperbolic ...
- 383
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4
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578
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Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?
Let $G$ be a finite group of diffeomorphisms of the torus $T^n$ fixing some point $p$, i.e. $p$ is fixed by every element of $G$. I have two questions.
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = ...
- 101
10
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1
answer
377
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Translation lengths in CAT(0) spaces
Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...
- 8,401
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4
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934
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Stallings' Theorem for free products of groups
There is a well known theorem which states that:
Theorem(Stallings):
For any immersion $f$ from a finite graph $D$ to $G$ there is a finite-sheeted covering space $D '$ of $G$ that extends $f$. ...
- 339
10
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1
answer
183
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The rigidity of the countable product of free groups
For a natural number $n$ let $F_n$ be the free group with $n$ generators.
The group $F_n$ is endowed with the discrete topology.
Given an increasing sequence $\vec p=(p_k)_{k\in\omega}$ of prime ...
- 41.8k
10
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2
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890
views
Are virtual cubulated groups cubulated?
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 complex?
Edit: After ...
- 539
10
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2
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815
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Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4
I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
- 201
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2
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631
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Invariant free factor of a free group
Let $F_n=F\ast F'$ be a free splitting of the free group $F_n$ and $\phi\in Aut(F_n)$. The free factor $F$ is said to be invariant under $\phi$ if $\phi(F)\subseteq F$.
I recently wondered if this ...
- 937
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1
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423
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Residually finite group surjective to nonresidually finite group with finitely generated kernel
As is described in the title, is there a known example such that there is a surjective homomorphism of groups $$f: G\rightarrow H,$$ with $G$ and $H$ finitely presented, $G$ is residually finite, and $...
- 497
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2
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724
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Centralizers of non-iwip elements of $Out(F_n)$
Does there exist an infinite order element $\phi\in Out(F_n)$, for some or all $n\geq 3$, which is not iwip but has finite index in its centralizer? How about an element such that all its non-zero ...
- 618
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1
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706
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Where to find English translation of Pansu's paper from Ann. Math?
Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...
- 28k
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2
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550
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Gromov hyperbolicity constant vs. Gromov-Hausdorff distance to a tree
Let $X$ be a compact, geodesic metric space which is Gromov hyperbolic with a constant $\delta>0$. To fix scaling, let us also assume that $X$ has diameter $1$.
To fix a definition of Gromov ...
- 101
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1
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602
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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^...
- 37.4k
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1
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434
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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 ...
- 1,090
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1
answer
1k
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CAT(0) groups that does not act on CAT(0) cubical complex
CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
- 1,598
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2
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454
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Minimal normally generating subsets of minimal generating sets
Let $G$ be a finitely generated group. The weight $w(G)$ of $G$ is defined to be the minimum number of elements of $G$ whose normal closure in $G$ is the whole of $G$ (this is sometimes also called ...
- 770
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788
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Are Hyperbolic Groups Residually Amenable
It is a well-known conjecture (or maybe just a question) that all hyperbolic groups are residually finite. What happens if we weaken the conclusion; in particular
Are all hyperbolic groups residually ...
- 2,441
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1
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737
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Parabolic subgroups of relatively hyperbolic and CAT(0) groups
Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...
- 2,090
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1
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459
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Kazhdan's property T for Kahler surfaces
Is it true that the fundamental groups of compact Kahler surfaces have property T if and only if it they are finite? I am having trouble finding counterexamples to this, but maybe that's just me...
- 96.4k
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2
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677
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Is every metric space quasi-isometric to a graph?
I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$...during this proof I've precisely used the fact that given two point in $X$ there ...
- 3,751
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1
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580
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Nonhyperbolic groups that contain no free abelian groups or Baumslag-Solitar groups
I've heard it conjectured that a finitely presentable group $G$ is hyperbolic if it satisfies the following two conditions.
$G$ contains no subgroup isomorphic to a Baumslag-Solitar group $BS(n,m)$ (...
- 103
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537
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Decidability of conjugacy problem for finitely generated subgroups of free groups
The conjugacy problem for a free group $F_n$ on $n$ letters has an easy solution. Each element of $F_n$ is conjugate to a unique and easily computable "cyclically reduced element" (this means that if ...
- 415
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2
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403
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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 $\...
- 949
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149
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Iterated algebraic fibering
A finitely generated group $G$ algebraically fibers if there is an epimorphism $G\to\mathbb{Z}$ with finitely generated kernel. Since this kernel is finitely generated, we can ask whether *it* ...
- 3,740
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416
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How are reflection groups related to general point groups?
I always tried to understand how the finite reflection groups of $\Bbb R^d$ (of some fixed dimension $d$) relate to the point groups of the same space $\smash{\Bbb R^d}$ (finite subgroup of the ...
- 13.6k
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383
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Can a positive measure subset of a free group be nowhere dense?
Let $F$ be a finitely generated free group and let $S \subseteq F$ be a subset for which there is some $\epsilon > 0$ such that for any epimorphism to a finite group $\phi \colon F \to G$ we have ...
- 11.3k
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223
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Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?
A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width.
Let $X$ be a geodesically complete CAT(0) ...
- 2,964
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351
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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 $A_{12} \in \Sigma_b$ ...
- 66.3k
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455
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Is there a one-relator circle-packing theorem?
Let $X_w$ be the presentation complex of a one-relator group $\langle x_1,\dotsc,x_n\mid w\rangle$, with $w$ cyclically reduced, i.e., $X_w=R\cup_w D$, with $R$ the rose with $n$ petals labeled $x_1,\...
- 436
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214
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2-generator subgroups of an Artin group of small type
Suppose I have an Artin group $G$ of small-type, meaning that the generators either commute or braid. E.g a braid group. Take two generators $g, h$ and arbitrary conjugates of these generators $xgx^{-...
- 765
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458
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is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?
This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...
- 101
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331
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A subgroup of corank 1 in a free group contains a primitive element?
Let $F$ be the free group on $\{x_i\}_{i=1}^\infty$, and let $H \leq F$ be a subgroup with $\langle H \cup \{x_1\} \rangle = F$. Must there be a free basis $B$ of $F$ for which $B \cap H \neq \...
- 11.3k
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3
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842
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Is there a one relator group with property (T)?
Is there a one-relator group with property (T)?
That is, is there an $n > 2$, and some $x \in F_n$ (the free group on $n$ generators) such that the quotient of $F_n$ by the normal subgroup ...
- 11.3k
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3
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1k
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Are subgroups of hyperbolic groups quasiisometrically embedded ?
Given a finitely generated subgroup of a finitely generated hyperbolic group. Is it true that the inclusion of each subgroup is a quasiisometric embedding ?
The first example for a group that does ...
- 11.1k
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1
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493
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Can $E_8$ be enlarged?
Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
- 2,773
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3
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498
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Residually solvable Bianchi groups
Let $d$ be a square-free positive integer, and let $\mathcal{O}_d$ be the ring of integers of the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$. Consider the Bianchi group $\Gamma_d = \...
9
votes
1
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467
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dichotomy in hyperbolic groups
Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...
- 301
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2
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396
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Conjugacy of matrices of order three in $PGL(2,k)$, where $k$ is any field
We know that every matrix of order three in $PGL(2,\mathbb Z)$ is conjugate to the following matrix
$$
\left(
\begin{array}[cc]
&1 & -1 \\
1&0
\end{array}
\right)
$$
I want to know if ...
- 91
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1
answer
674
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Can you decide whether the commutator subgroup of a f.p. group is f.g?
Is the following algorithmic problem known to be decidable/undecidable?
Input: a finite group presentation $P$.
Decide: is the commutator subgroup of the group presented by $P$ finitely generated?
- 343
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1
answer
398
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When are biautomatic groups hyperbolic?
This list of open problems from http://grouptheory.info/ includes the question:
"Is every biautomatic group which does not contain any $\mathbb{Z} \times \mathbb{Z}$ subgroups, hyperbolic?"
...
- 123
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1
answer
738
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Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
- 291
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votes
1
answer
377
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Morse theory on outer space via the lengths of finitely many conjugacy classes
Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...
- 93
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2
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762
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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 ...
- 9,237
9
votes
1
answer
308
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Counterexamples to analogue of Cannon conjecture in higher dimensions
It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for $\mathbb{H}^...
- 245
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votes
1
answer
371
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Finitely generated group every 2-generated subgroup of which is finite
I know of Tarski monsters and the Burnside Problem. I would like to know if there is an infinite finitely generated group $G$ such that for any $g$ and $h$ in $G$, the subgroup generated by $\{g,h\}$ ...
- 402
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votes
1
answer
281
views
Largest Hopfian quotient
Let $\Gamma$ be a group, say finitely generated if it helps. Does $\Gamma$ admit a largest Hopfian quotient? That is, does there exist a Hopfian quotient $H$ of $\Gamma$, such that every surjective ...
- 435
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votes
2
answers
373
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Is there a highly transitive action of a finitely generated torsion simple group?
Is there a highly transitive action of a finitely generated torsion simple group $G$ on $\mathbb{Z}$ ?
Highly transitive means $k$-transitive for each $k \in \mathbb{N}$, that is: for every two $k$-...
- 11.3k
9
votes
2
answers
954
views
Actions of Thompson group F
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 contain a binary tree.
...
- 4,705