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
949
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Distortion in the Brin-Thompson 2V
Is it known whether the Brin-Thompson 2V contains a distortion element? By this I mean an element $f$ such that the word norm $|f^n|$
grows sublinearly, and $f$ is of infinite order. If such an ...
8
votes
1
answer
153
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Are the non-free factors of Grushko decomposition of a finitely generated convex-cocompact (but not cocompact) subgroup of PSL$(2,\mathbb{R})$ finite?
See Grushko decomposition theorem.
Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?
...
3
votes
0
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371
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What about a Cayley n-complex for n>2?
Let $G$ be a finitely presented group. The Cayley graph of the finite generating set is a $1$-complex where the $0$-cells are the elements of $G$ and the $1$-cells are given by the generators (...
0
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0
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102
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Inverse limit in category of graphs
Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....
4
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0
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246
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Convex core and geometric finiteness of negatively curved manifolds
I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...
8
votes
2
answers
786
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Group of exponential growth always contains a free sub-group?
I am not very conversant with the growth of a group, so this may be a very silly question.
It is known that $F_2$, the free group of rank $2$, has exponential growth. I was wondering whether the ...
4
votes
0
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152
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Ends of a negatively curved Riemannian manifold
Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...
7
votes
4
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571
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Is every virtually free group residually finite?
Question: Is every (finitely generated) virtually free group residually finite?
A well-known question asks whether every hyperbolic group is residually finite (Mladen Bestvina. Questions in geometric ...
3
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0
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50
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Decomposition about splitting of symmetric spaces of compact type
I get stuck in the following question:
Why does a locally symmetric space of compact type $M$ split locally irreducible components of dimension $\geq 2$ which are Einstein? In particular, why are all ...
5
votes
1
answer
370
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Is this semi-direct product residually finite?
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can help me find a way to check the residual finiteness of this group.
Consider the ...
31
votes
2
answers
1k
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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 ...
1
vote
1
answer
155
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Number of orbits for abelian group actions
Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite.
Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
2
votes
1
answer
214
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Parahoric subgroup over a local field
$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...
11
votes
1
answer
496
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Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?
Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-...
4
votes
1
answer
279
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Permuting subgroups with the same finite index
Suppose that we have a finitely generated residually finite group $G = \langle g_1,\ldots,g_r \rangle$ with polynomial growth. Let $H$ be a subgroup of $G$ with finite index $m$. Let $\phi$ be an ...
6
votes
1
answer
224
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Positive harmonic functions on nilpotent groups & Random walk on groups with a finite number of generators
I want to read the following papers in the English version which I could not find anywhere (the only papers I can get are the Russian versions). Kindly help me out.
Gregory A. Margulis, Positive ...
9
votes
3
answers
503
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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 ...
5
votes
2
answers
264
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If $K\rtimes \mathbb{Z}$ is a finitely generated group but $K$ isn't, must the fixed points of $1_\mathbb{Z}$ be a finitely generated group?
I've copied over this question from what I asked on StackExchange, in the hope that an expert here can readily answer the question.
Is there an example of a group $G=K\rtimes \mathbb{Z}$ satisfying ...
7
votes
1
answer
136
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Discrete cocompact group of isometries of Nil
Is it true that every group quasi-isometric to the Heisenberg group admits a proper cocompact action by isometries on Nil?
5
votes
1
answer
145
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Given a quasi-convex subgroup $H$ of hyperbolic $G$, can we decide if two elements $x,y \in G$ lie in the same double coset of $H$?
I've come across the following question in my research, which seems elusive but is almost surely decidable.
Let $H$ be a quasi-convex subgroup of the hyperbolic group $G$. Given $x, y \in G$, we wish ...
5
votes
1
answer
91
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Infinite groups that admit a discrete, co-compact, bilipschitz action on $\mathbb{R}^3$
Apart from the abstract types of the crystallographic groups, are there any other abstract groups that admit a proper, co-compact, uniformly bilipschitz action on $\mathbb{R}^3$?
3
votes
0
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75
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First Betti number of finitely non-co-hopfian groups
Let $G$ be a finitely generated group. Assume that $G$ is a finitely non-co-hopfian group in the sense that there is a group embedding $i: G\hookrightarrow G$ such that $1<[G\colon i(G)]<\infty$...
5
votes
2
answers
794
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Potential counterexamples to Bass' trace conjecture
Motivation: The following is a theorem of Berrick-Hesselholt (essentially also due to Linnell, though not in this form):
Let $G$ be a group. Suppose that for every subgroup of $G$ isomorphic to $\...
4
votes
1
answer
146
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Are geometric actions on CAT(0) spaces with isolated flats minimal on the boundary?
Suppose $X$ is a $CAT(0)$ space with isolated flats, $\partial X$ its visual boundary and $G$ acts properly discontinuously amd cocompactly on $X$. Must the $G$ action on $\partial X$ have a dense ...
0
votes
1
answer
190
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Torsion-free subgroup of affine group
Let $G$ be a finitely generated group and $\varphi:G\to \operatorname{Aut}(\mathbb C)$ a homomorphism, where $\operatorname{Aut}(\mathbb C)$ is the group of complex affine transfromations $a z+b$.
Can ...
5
votes
0
answers
110
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Finitely presentable group with purely infinite full group $C^*$-algebra?
Does there exist an example of a finitely presentable group whose full group $C^*$-algebra is purely infinite,
resp. is it known to be impossible?
8
votes
1
answer
201
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Non-finitely presented FP groups with cohomological dimension $2$
In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a ...
1
vote
1
answer
231
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Which properties can be read off the balls of a Cayley graph?
For which properties (P) [of groups] does the following hold:
given a group $G$ which has a finite presentation with at most $n$ relations of length at most $\ell$, there is a $R(n,\ell)$ so that, if ...
4
votes
0
answers
185
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Normal subgroups of non-co-hopfian groups (example)
We call a group $G$ a non-co-hopfian group, if $G$ admits a non-isomorphic injective group homomorphism from $G$ to itself $i\colon G\hookrightarrow G$.
Consider a finitely presented non-co-hopfian ...
7
votes
2
answers
540
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Status of the Hopf-Thurston sign conjecture in dimension 4
A famous conjecture in topology asserts:
The Euler characteristic of a closed aspherical $2n$-manifold $M$ satisfies $(-1)^n\chi(M) \geq 0$.
This was conjectured by Hopf for manifolds with non-...
1
vote
1
answer
174
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Liouville property of hyperbolic spaces
It seems classically known (and mentioned in several papers without reference) that there exist bounded non-constant harmonic functions on the hyperbolic space $\mathbb{H}^n, n \geq 2$. I am ...
19
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0
answers
420
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Are braid groups known to not be linear over $\mathbb{Z}$?
$\DeclareMathOperator\GL{GL}$It is known that every braid group $B_n$ embeds as a subgroup of $\GL_m(\mathbb{Z}[q^{\pm 1},t^{\pm 1}])$, where $m=n(n-1)/2$ (see Krammer - Braid groups are linear). This ...
3
votes
0
answers
123
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Salvetti complexes and cohomology of affine completion of Artin groups $E_6$ and $E_7$
After the solution of the Brieskorn-Arnold Pham conjecture on the asphericity of a space for affine Artin groups by Paolini and Salvetti MR4243019 (arXiv), I would like to know if there are ...
14
votes
2
answers
868
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Quasi-isometry groups of metric spaces
Given a metric space $(X, d)$, we can consider the set of all quasi-isometries $f: X \to X$, and quotient out by the equivalence relation identifying $f$ and $g$ if $\sup_{x \in X}d(f(x), g(x))$ is ...
3
votes
0
answers
114
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Why inherit the Tits systems structure by a $B$-adapted homomorphism?
Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...
4
votes
0
answers
132
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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 $\...
4
votes
0
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145
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Polynomial invariants of infinite reflection groups
It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...
8
votes
1
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192
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For which planar topological spaces $Z$ does there exist a hyperbolic group $\Gamma$ with $\partial \Gamma \cong Z$?
Recall a topological space is called planar if it can be embedded in $S^2$. I'm interested in understanding hyperbolic groups with planar boundaries.
In [1], it is shown that if a one-ended hyperbolic ...
5
votes
1
answer
163
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Growth of the word norm for elementary matrices in $\rm SL_3 (\mathbb{Z})$
This is a reference request, since the answer is probably well known, but I could not find it.
Given a finitely generated group $\Gamma$ with a generating set $S$, define the word norm $l = l_S : \...
1
vote
1
answer
58
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Countability of the wobbling group of a bounded geometry metric space
Let $(X, d)$ be a uniformly discrete metric space of bounded geometry, that is, $\sup_{x \in X} |B_r(x)| < \infty$ for every $r \geq 0$ and there is a uniform $\delta > 0$ such that $d(x, y) \...
1
vote
1
answer
79
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Deduce unsolvability of $\operatorname{IP}(G_0)$ from the Adian–Rabin Theorem
$\operatorname{IP}(G_0)$: the special isomorphism problem for $G_0$, i.e., given $G_0$, determine if $G$ is isomorphic to $G_0$. My question is that how can we deduce from the Adian–Rabin theorem that ...
4
votes
2
answers
324
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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,\...
2
votes
0
answers
106
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Moment of the hitting measure of a subgroup
Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
3
votes
2
answers
324
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Hyperbolic volume of hyperbolic knots
Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...
8
votes
1
answer
334
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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?"
...
4
votes
1
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198
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Reference for Chebyshev centers
Today, I came across the concept of Chebyshev center twice.
In particular, it is the key tool in the very elegant paper "A fixed point theorem for $L^1$ spaces" by Bader, Gelander and Monod.
...
4
votes
0
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207
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Graphs with high girth and low diameter
As the title says, I'm interested in graphs with high girth and low diameter.
Given a class $\Gamma$ of finite $k$-regular graphs, call a $\Gamma$-graph GD-extremal if every $\Gamma$-graph either has ...
6
votes
1
answer
435
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Maximal symmetries
The aim of this question is to investigate how topological group actions on manifolds differ from more rigid actions (like smooth ones).
Let $M$ be a connected and second-countable manifold, and $d,d'$...
4
votes
0
answers
107
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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 ...
4
votes
1
answer
102
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Shortcutting quasigeodesics
Let $\Gamma$ be a connected graph, let $\lambda \ge 1$ and $c \ge 0$ be some constants. Recall that a combinatorial path $p$ in $\Gamma$ is said to be $(\lambda,c)$-quasigeodesic if for every ...