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
27
votes
4
answers
2k
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Units in the group ring over fours group after Gardam
Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$.
This conjecture asserted that if $\Gamma$ is a ...
- 6,652
20
votes
7
answers
5k
views
Understanding groups that are not linear
I have a really hard time "feeling" what it means for a group to fail to be linear. Vaguely, I'd like to know how one should think about such groups. More precisely:
What are some interesting ...
- 397
15
votes
3
answers
2k
views
Folner sets and balls
Several related questions were asked before on MO, but it is not clear to me if the following was settled.
Given a finitely generated amenable group, is it always possible to find some finite ...
- 992
10
votes
1
answer
416
views
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
10
votes
1
answer
1k
views
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
52
votes
14
answers
14k
views
Introductory text on geometric group theory?
Can someone indicate me a good introductory text on geometric group theory?
26
votes
1
answer
1k
views
Group with finite outer automorphism group and large center
Does there exist a finitely generated group $G$ with outer automorphism group $\mathrm{Out}(G)$ finite, whose center contains infinitely many elements of order $p$ for some prime $p$?
A motivation is ...
- 63.9k
23
votes
9
answers
9k
views
The free group $F_2$ has index 12 in SL(2,$\mathbb{Z}$)
Is there someone who can give me some hints/references to the proof of this fact?
- 1,702
18
votes
1
answer
849
views
Is Hopf property a quasi-isometry invariant?
Recall that a group $G$ is called Hopfian if every surjective endomorphism $G\to G$ is injective. Malcev observed that all finitely-generated (f.g.) residually finite groups are Hopfian. It is well-...
- 31.2k
12
votes
1
answer
603
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 ...
- 4,484
8
votes
2
answers
571
views
Is residual finiteness a property of "many" finitely presented groups?
Is there a reasonable random model for selecting a finitely presented group $G$ such that with positive probablity (or even with probability almost $1$) some of the following properties hold:
$G$ is ...
- 11.3k
8
votes
2
answers
343
views
Cubic almost-vertex-transitive graphs with given spanning tree
Consider the infinite 3-regular tree. Pick a vertex $C$, the "center".
For any integer $L\ge 1$ consider the closed ball, in the graph distance, of radius $L$ around $C$. Let $T_L$ be the induced ...
8
votes
2
answers
682
views
Ends of finitely generated torsion groups
It is known that the number of ends of a finitely generated group is 0,1, 2 or $\infty$.
Problem 1. What is known about the number of ends of infinite finitely generated torsion groups?
In ...
- 41.8k
7
votes
2
answers
2k
views
Residual Finiteness of Fundamental Group of Compact 3-Manifold
Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi_1(M)$ is residually finite. The outline of the proof is basically:
Reduce ...
- 4,425
3
votes
1
answer
171
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 ...
- 97
2
votes
1
answer
257
views
Engulfing Kleinian groups?
Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely
generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ?
I know that this is true for Fuchsian ...
- 11.3k
33
votes
1
answer
1k
views
Is this conjecture strictly weaker than P=NP?
My three computability questions are related to the following group theory question (first asked by Bridson in 1996):
For which real $\alpha\ge 2$ the function $n^\alpha$ is equivalent to the Dehn ...
user6976
29
votes
4
answers
2k
views
Trees in groups of exponential growth
Question: Let $G$ be a finitely generated group with exponential growth.
Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree?
...
- 25.5k
24
votes
4
answers
2k
views
Free splittings of one-relator groups
Roughly speaking, I want to know whether one-relator groups only have 'obvious' free splittings.
Consider a one-relator group $G=F/\langle\langle r\rangle\rangle$, where $F$ is a free group. Is it ...
- 25k
24
votes
1
answer
968
views
Groups whose finite index subgroups of fixed index are isomorphic
I am interested in finitely generated groups $G$ that are residually finite and have the following property: For each $d \geq 1$, $G$ has subgroups of finite index $d$, and all such subgroups are ...
- 11.9k
21
votes
4
answers
2k
views
Is there a non-Hopfian lacunary hyperbolic group?
The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...
- 25k
21
votes
1
answer
831
views
Can a hyperbolic, one ended, one relator group, have a shorter trivial word?
Let $G= \langle S \mid r \rangle$ be a one-relator presentation for a one-ended hyperbolic group, with $r$ cyclically reduced.
Question: Can there be a nontrivial word $w(S)$ which is trivial in the ...
user35370
20
votes
4
answers
2k
views
Cayley graph of $A_5$ with generators $(1,2,3,4,5),(1,4,3,2,5)$
The Cayley graph of $A_5$ with two generators of order 5 seems rather complicated. What is its graph genus (orientable or non-orientable)?
The best I could get by trial and error is an embedding ...
- 24.8k
19
votes
2
answers
5k
views
Minimal number of generators for $GL(n,\mathbb{Z})$
$\DeclareMathOperator{\gl}{GL}\DeclareMathOperator{\sl}{SL}$From de la Harpe's book "Topics in Geometric Group Theory" I learnt that $\gl(n,\mathbb{Z})$ is generated by the matrices $$s_1 = \begin{...
- 1,312
18
votes
4
answers
2k
views
Braid groups acting on CAT(0)-complexes
Does the braid group $B_n, n\ge 3$, act properly by isometries on a CAT(0) cube complex?
Update 1. During a recent talk of Nigel Higson in Pennstate Dmitri Burago asked whether the braid groups are ...
user6976
18
votes
1
answer
400
views
Finitely generated groups with Hölder-exotic space of ends?
The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...
- 63.9k
17
votes
3
answers
1k
views
Examples of locally hyperbolic groups
It is well-known that a subgroup of a hyperbolic group need not be hyperbolic. Let us say that a (finitely generated) group $G$ is locally hyperbolic if all its finitely generated subgroups are (...
- 383
16
votes
1
answer
916
views
Is it true that every f.g. infinite simple group has exponential growth?
Is it true that every finitely generated infinite simple group has
exponential (word-)growth?
Remark: As Mark Sapir has pointed out, the question whether
every finitely generated group of ...
- 19.6k
16
votes
2
answers
3k
views
The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
- 20.9k
15
votes
2
answers
886
views
Is there a finitely generated residually finite group with solvable word problem that does not embed in a finitely presented residually finite group?
The famous Higman embedding theorem says that every recursively presented group embeds in a finitely presented group. This is a convenient tool to construct finitely presented groups with bizarre ...
- 38.6k
15
votes
2
answers
2k
views
Dehn's algorithm for word problem for surface groups
For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...
- 153
13
votes
2
answers
498
views
Decidability of word problem for group admitting certain action
Let $G$ be a group acting highly transitively (and faithfully) on a set $S$. Suppose that $G$ is finitely presented, and that every stabilizer in $G$ of a finite subset of $S$ is finitely generated. I ...
- 3,740
13
votes
1
answer
1k
views
Topology of boundaries of hyperbolic groups
For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...
- 305
13
votes
1
answer
910
views
Holomorphic cusp forms and cohomology of GL(2,Z)
Let $V_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute ...
- 40.2k
12
votes
1
answer
462
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. ...
- 949
12
votes
0
answers
373
views
Does Thompson's group $V$ have property AP?
Property AP: A discrete group $\Gamma$ has property AP (Approximation Property) if there exists a net $(\phi_i)_{i \in I}$ of finitely supported functions on $\Gamma$ such that $\phi_i \to 1 $ weak$^*$...
11
votes
2
answers
755
views
Quasi-isometric rigidity of Nil
Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
- 3,749
11
votes
1
answer
598
views
If $(F_n)_n$ is a Følner sequence satisfying Tempelman's condition, is $(F_n^{-1}F_n)_n$ also Følner?
Let $G$ be a countable group. A Følner sequence is a sequence of finite subsets $(F_n)_n$ such that
$$\lim_{n\to\infty} \frac{|KF_n \mathbin\triangle F_n|}{|F_n|} = 0$$
for each fixed finite subset $K ...
- 213
10
votes
2
answers
815
views
Paper by I. N. Sanov, Solution of the Burnside problem for exponent 4
I have searched extensively online and for copies of printed journals containing the paper which details Sanov's solution to the Burnside Problem for exponent 4, which is widely cited in many papers ...
- 201
10
votes
2
answers
2k
views
When is a Baumslag-Solitar group linear?
The Baumslag-Solitar group $BS(m,n)$ is given by the group presentation
$BS(m,n)=(a,b|ba^{m}b^{-1}=a^{n})$. When does it embed into a linear group? Thanks!
- 411
9
votes
1
answer
308
views
Counterexamples to analogue of Cannon conjecture in higher dimensions
It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for $\mathbb{H}^...
- 245
9
votes
1
answer
493
views
Can $E_8$ be enlarged?
Is there any finite 8-dimensional point group which contains the $E_8$ Coxeter group as a subgroup other than $E_8$ itself?
- 2,773
9
votes
4
answers
982
views
isometric embeddings of Cayley graphs in "nice" spaces
This is from a physicist I know and as may be expected, I am threading my way between poorly defined and poorly translated.
What groups have Cayley graphs (w.r.t. a fixed finite generating set, and ...
- 1,625
9
votes
0
answers
310
views
Breuer-Guralnick-Kantor conjecture and infinite 3/2-generated groups
A group $G$ is called $\frac{3}{2}$-generated if every non-trivial element is contained in a generating pair, i.e. $$\forall g \in G \setminus \{e \}, \ \exists g' \in G \text{ such that } \langle g,g'...
9
votes
3
answers
842
views
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
8
votes
1
answer
362
views
If a group $G$ has decidable word problem, must it have a decidable square problem?
My question is a refinement of this one about 'efficient' construction of square elements: If the word problem for a (finitely generated, finitely presented) group is decidable, must the 'square ...
- 1,951
8
votes
0
answers
185
views
Sharp isoperimetry in the discrete Heisenberg group
The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...
- 4,432
7
votes
3
answers
523
views
Membership to double cosets in free groups
Is there an elementary and efficient algorithm for testing the membership to a double coset of f.g. subgroups in a free group?
Has this membership problem been implemented in GAP/Magma?
More ...
- 3,181
7
votes
1
answer
319
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}...
- 6,652
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