All Questions
Tagged with gr.group-theory mg.metric-geometry
36
questions with no upvoted or accepted answers
14
votes
0
answers
544
views
Who conjectured that a transitive projective plane is Desarguesian?
The only known finite projective plane with a transitive automorphism group is the Desarguesian plane $PG(2,q)$ and it seems likely that there are no others, although this is not (quite) proved.
...
- 11.8k
14
votes
0
answers
681
views
Are all these groups CAT(0) groups?
Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive ...
- 10.1k
11
votes
0
answers
659
views
Uniquely geodesic groups
Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...
- 25.5k
10
votes
0
answers
364
views
Kissing the Monster, or $196,560$ vs. $196,883$
The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...
- 209
10
votes
0
answers
208
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) ...
- 2,481
10
votes
0
answers
449
views
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
8
votes
0
answers
429
views
A lattice with Monster group symmetries
The book Mathematical Evolutions contains the following excerpt:
A last, famous, example is the following. It is known that in the space
of one hundred and ninety six thousand eight hundred and ...
- 12.1k
8
votes
0
answers
174
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 ...
- 4,342
7
votes
0
answers
306
views
Status of an open question in Artin's "Geometric Algebra"
In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2).
The ...
- 501
7
votes
0
answers
153
views
Connectedness of cones in the boundary of a 1-ended hyperbolic group
Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
- 611
7
votes
0
answers
312
views
Erlangen program carried out explicitely?
I'm looking for a book where the Erlangen program is carried out on some example groups with explicit computations.
What I mean by "carrying out Erlangen program" is picking a specific group (say SO(...
- 71
5
votes
0
answers
135
views
Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?
$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
- 51
5
votes
0
answers
156
views
Subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$
Is there a classification theorem for the subgroups of $\mathrm{O}_3$ that are the symmetry groups of compact subsets of $\mathbb{R}^3$?
Apparently, there is an almost complete classification in ...
- 6,164
5
votes
0
answers
142
views
Uniform versus non-uniform group stability
Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, ...
- 939
5
votes
0
answers
168
views
Non-algebraic quasi-isometric embeddings
What are examples of finitely generated groups $\Gamma$ and $\Lambda$ such that the metric space $\Lambda$ embeds into $\Gamma$ quasi-isometrically but such that $\Lambda$ is very much not a subgroup ...
- 2,030
5
votes
0
answers
229
views
Uniqueness of the boundary of a hierarchically hyperbolic group
Hierarchically hyperbolic groups and spaces (HHG and HHS for short) were defined by Behrstock, Hagen and Sisto (see here and here). Examples include mapping class groups, Right angled Artin groups, ...
- 1,835
5
votes
0
answers
134
views
Filling points to a simplex in models for EG
I have a question which is related to higher Dehn functions of groups.
I also have a group $G$ with a finite $K(G,1)$. Let us denote by $EG$ the universal cover of this complex. We choose a path-...
- 2,896
5
votes
0
answers
195
views
Coarsely Lipschitz retractions onto cyclic subgroups
A good way to show that a subspace is undistorted is to give a coarse Lipschitz retraction of the whole space onto that subspace. This question is about a failure of the converse.
Let $G$ be a ...
4
votes
0
answers
124
views
Electrifications of quasi-geodesics in relatively hyperbolic groups
This post is somewhat of a followup to my previous post here. $\DeclareMathOperator\Cay{Cay}$Suppose $G$ is a relatively hyperbolic group with peripheral subgroups $P_1,P_2,\dots, P_n$, and suppose $\...
- 379
4
votes
0
answers
103
views
Sufficient conditions for the Besicovitch covering theorem to hold on groups of polynomial growth
Let $G$ be a finitely generated group with symmetric generating set $S$. Then $S$ induces a distance $d$ on $G$ by letting $d(a,b) = $ the minimum $n$ such that there are generators $s_1,...,s_n$ with ...
- 41
4
votes
0
answers
104
views
Hilbert space compression of lamplighter over lamplighter groups
$C_2 \wr \mathbb{Z}$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space.
Question: Consider the group $C_2 \wr (C_2 \wr \mathbb{Z})$, what is ...
- 4,342
3
votes
0
answers
135
views
Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?
It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...
- 460
3
votes
0
answers
224
views
Clarifications involving automorphisms of projective planes and lines?
I have been learning some classical projective geometry recently and I am hoping to gain some clarity regarding various different automorphism groups. There are three different levels of generality ...
- 1,045
3
votes
0
answers
137
views
Erlangen program for "network geometry"
The subject of network geometry (Boguna et al., Network Geometry,
Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks.
This is about studying a metric on the nodes, ...
- 612
3
votes
0
answers
103
views
Degree of a local cut point in the boundary of a hyperbolic group
Suppose $G$ is a one-ended word-hyperbolic group and $\xi$ is a (local) cut point of $\partial G$. Fix any visual metric on $\partial G$ and let $U(\epsilon,\xi)$ be the connected component of $\xi$ ...
- 611
3
votes
0
answers
152
views
Representing discrete groups in orthogonal groups
Suppose that we have a matrix $A$ of a quadratic form $Q_A$ of signature $(n,1)$ and a matrix $B$ of a quadratic form $Q_B$ which also has signature $(n,1)$. Let $O(Q_A)$ be the orthogonal group that ...
- 41
3
votes
0
answers
464
views
Higher order Pansu derivative
Given a group $(G,*)$ there is no candidate for what can be understood as a derivative of a function $$f:G\rightarrow\mathbb{R}.$$ However, for the special case of Carnot groups there is the so-...
- 31
2
votes
0
answers
220
views
Discs of minimal area in CAT(0) spaces
Let $X$ be a CAT(0) space, $\gamma$ a closed curve in $X$ of length $n$. Let $D$ be a disc with boundary $\gamma$ of smallest possible area. Suppose that the area of $D$ is bigger than $cn^2$ for some ...
user6976
2
votes
0
answers
238
views
Finitely generated groups non-embeddable into $L_1(0,1)$
I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group $\...
- 4,858
1
vote
0
answers
54
views
Neighbor count in sphere packing in N dimensions
So I'm really interested in building a mathematical model for how powerful computer chips could be given extra spatial dimensions. Obviously this is a squishy problem, since "computer chips" ...
- 111
1
vote
0
answers
44
views
How dense can a transitive sets of points be?
How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...
- 11.9k
1
vote
0
answers
42
views
In 3D point groups, does $[\Gamma_{e}\otimes\Gamma_e] = \Gamma_{Rot_z} \forall$ degenerate $\Gamma_e$ hold in general?
In the following I am referring to groups exclusively describing 3D point symmetries. I use the Schönflies notation for groups and their elements and the Mulliken symbols to describe their irreducible ...
1
vote
0
answers
121
views
Asymptotic cone of discrete group of Heisenberg group $\mathbb{H}^3$
Note that $(\mathbb{Z}^2,d_W)$ where $d_W$ is word metric has asymptotic cone $$(\mathbb{R}^2,\| \ \|_1)=\lim_{t>0\rightarrow 0}\ t(\mathbb{Z}^2,d_W)$$
And Heisenberg group $\mathbb{H}^3$ has an ...
- 1,058
1
vote
0
answers
249
views
Virtually abelian centralizers
This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.)
Anyway, I'm wondering what sort of groups have the ...
- 237
0
votes
0
answers
294
views
Isometry group of a complete separable metric space is Polish?
Let $(X,d)$ be a complete separable metric space, and endow $Iso(X,d)$ with the pointwise convergence topology.
I've seen a few sources say this is clearly a Polish group, but why is this this the ...
- 568
0
votes
0
answers
1k
views
Bi invariant Riemannian metric on a Lie Group
I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...
