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
951
questions
13
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3
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2k
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Which groups are LERF?
A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...
- 3,171
11
votes
3
answers
344
views
Right-angled Artin groups that split as direct products
For a finite graph $X$, let $A_X$ denote the associated right-angled Artin group. Thus $A_X$ is generated by the vertices of $X$ subject to the relations $[v,w]=1$ whenever vertices $v$ and $w$ are ...
- 7,511
5
votes
0
answers
373
views
Hyperbolic groups and residual finiteness
The existence of a hyperbolic group which is not residually finite is (to my knowledge) an open question. Is there any reason to suspect that all hyperbolic groups are residually finite, perhaps some ...
- 601
12
votes
0
answers
459
views
Writing an element of a free product of $C_2$'s as a product of order-$2$ elements
My question is simple: Suppose that $G$ is isomorphic to the free product of finitely many copies of $C_2$. Is it true that any element $g \in G$ can be written as a product $g = s_1 \dotsm s_m$ such ...
- 1,308
1
vote
1
answer
176
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A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
- 322
3
votes
1
answer
187
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Maps of surfaces to CAT(0) cube complexes
Let $C$ be a locally CAT(0) cube complex. Let $f\colon \Sigma_g \rightarrow C$ be a continuous map from a closed oriented genus $g \geq 1$ surface to $C$. Is it always possible to find a cube ...
- 26.4k
16
votes
0
answers
337
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Does every infinite, connected, locally finite, vertex-transitive graph have a leafless spanning tree?
My question is
Let $G$ be an infinite, connected, locally finite, vertex-transitive graph. Must
$G$ have the following substructures?
i) a leafless spanning
tree;
ii) a spanning forest consisting ...
- 22.9k
4
votes
1
answer
182
views
Groups that don't contain quasi-hyperbolic plane
Is there any known example of a one-ended finitely presented group with exponential growth that does not contain a quasi-isometric copy of the hyperbolic plane?
This question is motivated by the ...
- 60.4k
6
votes
2
answers
190
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Are the canonical embeddings into $G*_AH$ quasi-isometric?
Suppose $A,G,H$ are finitely generated groups and $A$ is quasi-isometrically embedded into $G$ and $H$. Does it follow that the natural embeddings of $G$ and $H$ into $G*_AH$ are quasi-isometric?
I ...
- 9,748
3
votes
0
answers
106
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Finite homology of a homogeneous space
Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be ...
- 60.4k
31
votes
2
answers
1k
views
Group theory with grep?
While reading Bill Thurston's obituary in the Notices of the AMS I came across the following fascinating anecdote (pg. 32):
Bill’s enthusiasm during the early stages of mathematical discovery was ...
- 737
7
votes
0
answers
235
views
In search of a quick proof that groups acting freely on $\mathbb R$-trees are linear
Many years ago I had the idea to use non-standard analysis to prove that a group acting freely on an $\mathbb R$-tree must be linear. The heuristic went like this:
A non-standard model $G^*$
of the ...
- 4,210
6
votes
0
answers
158
views
Examples of groups with a positive homogeneous presentation without the Haagerup property or not of type $F_\infty$
I am looking for groups with a certain presentation that do not have the Haagerup property or are finitely presented but not of type $F_\infty$ (meaninig that for some $n\geq 3$ we cannot find any ...
- 61
2
votes
0
answers
117
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Convex subsets in abstract groups
Consider a group $G$. A norm on $G$ is a function $\|\cdot\|\colon G\to\mathbb R_+$ with (*) $\|gh\|\le\|g\|+\|h\|$ and $\|g\|=\|g^{-1}\|$ and $\|1\|=0$; the space of norms is a convex in $\mathbb R_+^...
- 2,489
12
votes
1
answer
380
views
What is the least $n\ge1$ for which there is an $n$-dimensional closed flat manifold with perfect fundamental group?
In answer to the question "Is there a flat manifold with trivial first homology?" I proposed choosing a finite perfect group $P$ and a surjection $\phi:F\to P$ where $F$ is a free group of ...
- 1,446
9
votes
1
answer
215
views
Yang-Mills algebra and lower central series of surface groups
Here is a connection that I recently noticed, but I haven't quite been able to make sense of. It might follow from well-known facts; apologies, if so. This is quite far from my area.
First, in "...
- 8,113
5
votes
1
answer
480
views
What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\...
- 11.4k
3
votes
0
answers
88
views
Reference for Varopoulos isoperimetric inequality with multiplicity
The Varopoulos isoperimetric inequality for a bounded domain $D$ in a nilpotent group $\Gamma$ of growth $n$ reads
$$
\# D \le \mathrm{const} \cdot (\#\partial D)^{n/(n-1)}
$$
See Ch. 6.E+ in Gromov's ...
- 60.4k
2
votes
1
answer
66
views
Sets with a good lift under a covering
Suppose I have a covering map $\pi : E \to X$ between (nice) topological spaces, and $x \in X$. If $U \ni x$ is a very small open set, then $\pi^{-1}(U)$ is a discrete union of subsets $V_d \subset X$ ...
- 5,016
2
votes
0
answers
166
views
Characterization of growth in terms of coarse algebraic topology
$$
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mbb}[1]{\mathbb{#1}}
\newcommand{\opn}[1]{\operatorname{#1}}
\DeclareMathOperator\cap{cap}
\def\sse{\subseteq}
$$
Coarse spaces
Let $X$ be a coarse ...
- 1,294
6
votes
1
answer
150
views
Does the visual boundary of any one-ended Cayley graph contain at least three points?
Let $\Gamma$ be a Cayley graph of a finitely generated group. We can define the visual boundary of $\Gamma$ with respect to some base vertex $b$, denoted $\partial \Gamma$, as the set of geodesic rays ...
- 60.4k
4
votes
0
answers
140
views
A variation of Zuk's isoperimetric inequality for groups
$\DeclareMathOperator\diam{diam}\DeclareMathOperator\inrad{inrad}$There is a isoperimetric inequality (conjectured by Sikorav and proven by Żuk (Topology 39 (2000) 947–956) which holds in every Cayley ...
- 4,342
7
votes
1
answer
224
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If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?
I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post.
I am reading this thesis.
Corollary 4.1.15. on page 63 ...
- 1,900
9
votes
3
answers
503
views
Spectral radius of a finitely generated group
Let $G$ be a finitely generated group and $\Gamma$ be its Cayley graph with the usual word metric. Let $\mu$ be a symmetric non-degenerate measure on $G$ (maybe with finite support or smooth), and ...
- 1,900
1
vote
1
answer
172
views
What properties are preserved by quasi-isometries
Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones".
What (metric)/geometric properties are ...
- 415
3
votes
1
answer
301
views
Can graphs of groups be thought of as "graph objects" in the category of groupoids?
An undirected graph is sometimes defined as a pair of sets $V$ and $E$ (vertices and oriented edges), together with two maps $i,f: E\to V$ (sending a directed edge its initial/final vertex) and a map $...
- 1,986
2
votes
1
answer
486
views
Growth rate of an outer automorphism of a free product
$\DeclareMathOperator\Out{Out}$Let $G=G_1\ast\cdots\ast G_k\ast F_p$ be a Grushko decomposition of a finitely generated group $G$, $\mathcal{O}$ the outer space relative to this decomposition, $[\phi]\...
- 1,986
10
votes
2
answers
523
views
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 ...
- 43.7k
9
votes
4
answers
1k
views
Proving that a countable group is not finitely generated
I would like to learn about techniques for proving that a countable group is not finitely generated. I am also interested in learning about examples. Finally, I am particularly, but not exclusively, ...
- 19.5k
17
votes
1
answer
361
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 ...
- 60.4k
10
votes
0
answers
214
views
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) ...
- 415
6
votes
0
answers
338
views
Arithmetic Teichmüller curves, first eigenvalue of the Laplacian, McMullen's expander conjecture
$\DeclareMathOperator\SL{SL}$By a result due to Ellenberg and McReynolds, any finite index subgroup $\Gamma$ of $\Gamma(2) \subset \SL\left(2,\mathbb{Z}\right)$ is the Veech group of an arithmetic ...
- 11
7
votes
1
answer
153
views
Density of “diagonal sets” in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
- 11.4k
1
vote
0
answers
69
views
Another matrices for a semigroup with intermediate growth
Nathanson showed that the Okninski's semigroup $S$ of $2×2$ matrices which is generated by the set $H=\{A,B\}$, where
$
A=\begin{bmatrix}
1&1\\
0&1\\
\end{bmatrix}
,
B=\begin{bmatrix}
1&0\\...
- 885
4
votes
1
answer
500
views
Amenable subsets of groups
Let $X$ be a subset of a group $G$. We say that $X$ is left amenable with respect to $G$ if there is a function $\mu:\mathcal P(G)\to [0,\infty]$ with the following three properties.
$\mu(A\cup B)=\...
- 257
2
votes
0
answers
144
views
"Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\CC{C}$In the article "Hyperbolic rigidity of higher rank lattices", Thomas Haettel has proved the following theorem: Let $\Gamma$ be a ...
- 60.4k
10
votes
1
answer
330
views
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 \...
CommunityBot
- 1
10
votes
1
answer
138
views
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* ...
- 3,431
4
votes
0
answers
74
views
Counting the number of free bases of $F_n$ with elements of bounded length
Let $F$ be a free group of finite rank and fix a free generating set $X$ of $F$. Let $P_r$ denote the set of all free generating sets of $F$ whose elements have length bounded by $r$ (when considered ...
- 250
5
votes
1
answer
306
views
In what sense is Bass-Serre theory the one-dimensional version of orbifold theory
The Wikipedia article on Bass-Serre theory claims that graphs of groups (in the context of Bass-Serre theory) "can be viewed as one dimensional versions of orbifolds." I hazily see a ...
- 115k
4
votes
1
answer
187
views
Infinitely divisible elements in Gromov hyperbolic groups
An element $g\in G$ in a group $G$ is called infinitely divisible if $b=y^n$ for infinitely many different $n\in {\Bbb Z}$. It is not hard to find a finite CW-complex (or even a compact manifold) ...
- 26.4k
2
votes
0
answers
101
views
Definition of the category QMet of metric spaces and quasi-isometries
I am following Clara Löh's Geometric Group Theory. An Introduction, and in remark 5.1.12, she defines the category QMet whose objects are metric spaces and whose morphisms are quasi-isometric ...
- 8,076
2
votes
1
answer
136
views
Subgroup growth of direct product
I have started reading about subgroup growth and, to my surprise, I haven't found a reference to whether direct products preserve subgroup growth.
Recall that, given a finitely generated group $G$, ...
- 5,200
0
votes
0
answers
58
views
How large must "weak Besicovitch" subsets of groups be?
Consider a group $G$; let call $A\subset G$ a weak Besicovitch subset whenever every element of $G$ can be written under the form $gh^{-1}$, where $g,h\in A$.
General question: how large must a weak ...
- 14.2k
7
votes
1
answer
464
views
Rational stable translation length
Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...
- 7,511
6
votes
2
answers
642
views
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 ...
- 7,511
3
votes
0
answers
95
views
Order type of monotone functions on $\Bbb N$ up to affine conjugation
Let's introduce order on non-strictly monotone functions $\Bbb N \to \Bbb N$ such that $f \leq g$ if $f(n) \leq Cg(Cn + C) + C$ and, of course, identify such $f, g$ if $f \leq g \leq f$. (Note absence ...
- 4,436
5
votes
3
answers
588
views
Does a compact semilocally simply connected geodesic space have the homotopy type of a CW complex?
Does a compact semilocally simply connected geodesic space have the homotopy type of a compact CW complex? Actually what I'd like to know is whether the fundamental group of such a space is finitely ...
- 1,596
3
votes
0
answers
126
views
the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
- 763
11
votes
1
answer
559
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 ...
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