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
82 questions
7
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Two-relator products of cyclic groups
In "A proof of the Scott–Wiegold conjecture on free products of cyclic groups" Howie proved that every one-relator product of three cyclic groups is nontrivial. Is there a now proven theorem that says ...
- 1,000
7
votes
2
answers
589
views
Dehn function for undistorted subgroups of a product of free groups
Let $G$ be a finitely generated subgroup of a product of two finite rank free groups $F_m \times F_n$. If there is a Lipschitz retraction $F_m \times F_n \to G$ with respect to word metrics, then $G$ ...
- 15.4k
6
votes
1
answer
193
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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 $\{...
- 6,652
6
votes
2
answers
575
views
Which groups are doubling?
A metric space $(M,d)$ is doubling if there exists $n$ such that every ball of radius $r$ can be covered by $n$ balls of radius $r/2$, for all $r$. For which f.g. groups $G$ and finite symmetric ...
- 6,652
6
votes
1
answer
406
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Connection between Stalling's end theorem and Seifert-van Kampen Theorem
Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
- 721
6
votes
0
answers
245
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Wild automorphisms of a free group
Let $F_X$ be a free group on a countably infinite set $X$. Let $\alpha$ be an automorphism of $F_X$ and $H$ a closed subgroup of $F_X$ in the profinite topology.
Is it possible that $\alpha(H) \...
- 11.3k
6
votes
1
answer
226
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Are $G$-limits of a slender group $G$ in the space of marked groups also slender?
A group $G$ is slender if every subgroup $H \leq G$ is finitely generated. This includes polycyclic-by-finite groups. Such groups are also called noetherian.
Suppose that $L$ is a $G$-limit group in ...
- 1,033
6
votes
2
answers
729
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Rationality of translation lengths in hyperbolic groups
Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...
- 619
5
votes
1
answer
138
views
Groups (not) quasi-retracting onto $\mathbb{Z}$
Let $X$ and $Y$ be metric spaces, with metrics $d_X$ and $d_Y$ respectively. A function $f\colon X\to Y$ is coarse Lipschitz if there exist constants $C,D>0$ such that $d_Y(f(x),f(x'))\le C d_X(x,x'...
- 3,740
5
votes
0
answers
169
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In the literature on infinite graphs, are there results on "periodizable" graphs?
Let $G=(V,E)$ be a connected countably infinite $k$-regular simple graph (no loops or multiple edges). For $A$ a finite subset of $V$, let me denote by $G_A=(A,E_A)$ the induced subgraph with vertex ...
5
votes
1
answer
342
views
Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...
- 2,090
5
votes
1
answer
242
views
Cancellation of elements in the Gromov boundary of a free group
Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
- 307
5
votes
1
answer
237
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Characterization of exact groups via the existence of amenable actions on unital C*-algebras, part 2
In this recent MOF question I asked whether exact groups could be characterized via the existence of amenable actions on unital C*-algebras. The answer, provided by Caleb Eckhardt in a comment, was ...
- 2,263
5
votes
0
answers
183
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Finitely generated nilpotent groups as cusp groups
I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
- 2,090
5
votes
0
answers
293
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Quotients of an extension of the Higman group
(Note: this started as a different question that soon changed form, thanks to the answers.)
Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ ...
- 20.2k
5
votes
3
answers
396
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Graphs of groups with homomorphisms not necessarily injective
I'm wondering if there is any literature on graphs of groups where the maps $G\to H$ from an edge group $G$ to its endpoint group $H$ are not necessarily $\pi_1$-injective. Or is this just too general ...
- 765
5
votes
1
answer
617
views
Fixed points on boundary of hyperbolic group
Let G be a word-hyperbolic group with torsion and let ∂G be its boundary. Do there exist criteria that imply that all non-trivial finite order elements of G act fixed-point freely on ∂G?
- 245
4
votes
1
answer
242
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Which 3-manifolds have positive rank gradient?
For which $3$-manifolds $M$ is the fundamental group $\pi_1(M)$
finitely generated and has positive rank gradient?
Recall that the rank gradient of a finitely generated group $G$ is defined to be $$...
- 11.3k
4
votes
3
answers
376
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Proof that lifts of geodesics are quasi-geodesics (relatively hyperbolic groups)
$\DeclareMathOperator\Cay{Cay}$Suppose $G$ is a relatively hyperbolic group with peripheral subgroups $P_1,P_2,\dots, P_n$, and suppose $\mathcal{S}$ is a finite generating set for $G$. Let $X=\Cay(G,\...
- 421
3
votes
1
answer
255
views
Group acting on a CAT(0) cube complex then acting also on a tree
If a group $G$ acts on a CAT(0) cube complex, then does $G$ act on a simplicial tree?
- 81
3
votes
1
answer
739
views
Bases of free groups
Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...
- 11.3k
3
votes
0
answers
414
views
Geometric intersection number for product of elements of the fundamental group
Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...
- 1,713
3
votes
1
answer
374
views
Is a finitely generated residually free group "almost LERF"?
Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
Let $...
- 11.3k
3
votes
1
answer
527
views
How do Dehn functions of special linear and mapping class groups behave?
Hi,
I apologize for the basic questions. I am looking for good references on the following problems:
1) What is known about the Dehn function of $SL_n(\mathbb{Z})$?
2) What is known about the Dehn ...
- 33
3
votes
1
answer
396
views
Are there quasiconvex normal subgroups?
Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?
- 11.3k
3
votes
1
answer
196
views
Non-cocompact action on CAT(0) cube complex and hyperplane stabilizers
A hyperplane of a cube complex $X$ is a connected component of taking an $(n-1)$-cube for each midcube of $X$ and identifiying midcubes along faces of adjacent $n$-cubes of $X$.
If a group $G$ acts ...
- 33
3
votes
0
answers
421
views
Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...
- 11.3k
2
votes
1
answer
439
views
Quotient groups of the lower central series of a surface group
In the answer to MO question 132247, it is possible to find a nice computation of the quotient groups of the lower central series of a finitely generated free group.
Q. What are the quotient ...
- 66.3k
2
votes
0
answers
147
views
Rank gradient in free products amalgamating a finite subgroup
Let $A,B$ be finitely generated groups with a common finite subgroup $C$. Suppose that $[A : C] > 2, [B : C] > 1$.
Must $A *_C B$ have positive rank gradient?
See Which 3-manifolds have ...
- 11.3k
2
votes
1
answer
192
views
Relation between Riemannian and Cayley-graph distance in a finite Coxeter group
Background: Let $W$ be a finite reflection group of rank $n$, acting on $\mathbb{R}^n$. The reflecting hyperplanes of $W$ meet the unit sphere $S^{n-1}\subset\mathbb{R}^n$, inducing a simplicial ...
- 2,851
2
votes
2
answers
757
views
Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?
Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups?
I know that $\mathrm{Aut}(\mathbb{Z}^n)\...
- 33
2
votes
1
answer
555
views
Kropholler's Conjecture and 3-manifolds
Suppose that $G$ is a group and that $H$ is a subgroup, both finitely generated, and assume that there is a non-trivial H-almost invariant set $X$ with $HXH=X$. Kropholler's Conjecture asserts that $ ...
