All Questions
Tagged with geometric-group-theory reference-request
89 questions
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The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
2
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1
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202
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Strong Liouville property of virtually abelian groups
Let $G$ be a finitely generated group and let $\mu$ be a symmetric non-degenerate measure on $G$. By strong Liouville property for $(G, \mu)$, we mean that every positive $\mu$-harmonic function on $G$...
- 1,407
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93
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References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections
A basic consequence of the Seifert–van Kampen theorem is the following.
Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-...
- 1,033
11
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250
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Recognising the elements of the Grigorchuk group
The Grigorchuk group $\mathfrak{G}= \langle a,b,c,d \rangle$ is a group of automorphisms of the infinite rooted binary tree $\mathcal{T}_2$. Every element of $\mathfrak{G}$ can be represented by a ...
- 8,401
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209
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A few points of clarification on the Martin boundary
Let $\Gamma$ be a finitely generated group, and let $M$ be the Martin boundary of $\Gamma$. I was reading the article on Martin boundary on Encyclopedia of Math, and I have a few questions about what ...
- 1,407
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2
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223
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Measures with superexponential moments on finitely generated groups
Let $\Gamma$ be an infinite finitely generated group and let $\nu$ be a measure on $\Gamma$ which generates a transient random walk. I was reading this paper, and the authors prove many of their ...
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Amplification argument for hyperlinear groups
Let us define a group $G$ to be a hyperlinear group if it satisfies the conclusion of Theorem 3.6. in these notes by Vladimir Pestov. It is well-known that one can use the so-called amplification ...
- 6,214
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When the fundamental group of subgraph of groups embeds?
Given a connected graph of groups $\mathcal G$ (where edge maps are embeddings), by a subgraph we mean a graph of groups obtain by omitting some vertices, some edges, and replacing the remaining ...
- 1,338
6
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1
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207
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
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202
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Green's kernel estimates on finitely generated groups
I was reading a paper by W. Hebisch and L. Saloff-Coste titled "Gaussian Estimates for Markov Chains and Random Walks on Groups" where I came to know about certain bounds on convolution ...
- 131
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Virtual fibring of $\mathrm{Out}(F_2\times F_2)$
A finitely generated group $G$ is said to virtually fibre if there is a finite index subgroup $H\leq G$ and a non-trivial map $\varphi:H\to\mathbb{Z}$ with $\ker(\varphi)$ finitely generated.
I want ...
- 911
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Does every amenable group $G$ admit a two-sided Folner sequence?
By two-sided Følner sequence I mean a sequence $(F_N)_N$ of subsets of $G$ which is both a left-Følner and a right-Følner sequence.
Context: I just came up with this question and surprisingly I haven'...
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155
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Variants of the Bonk-Schramm embedding
Recently I heard about the following embedding theorem of Bonk and Schramm: every Gromov hyperbolic geodesic metric space with "bounded growth" is roughly similar to a convex subset of $\...
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2
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311
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Injective hulls of metric spaces
In the context of large scale geometry and geometric group theory, I have recently come across the concept of injective hulls of metric spaces. For a metric space $X$, let $\text{In}(X)$ be the set of ...
- 41
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Strong converse of Kazhdan's property (T)
In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(...
- 1,027
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150
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$K(\pi,1)$-conjecture ofr Artin groups behave well with respect to special subgroups. Reference-Request
For a proof for an article I would need the following result:
If $A_\Gamma$ is an Artin group such that the $K(\pi,1)$-conjecture holds for it and $\Gamma'\subset\Gamma$ is an induced subgraph, then ...
- 911
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92
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Property A, Higson-Roe condition and its applications
Recently I have been studying amenability of groups and property A, and I came across the Higson-Roe condition:
Let $X$ be a uniformly discrete metric space with bounded geometry. $X$ has property $A$ ...
- 21
3
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1
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271
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Passing to normal forms in graphs of groups
Given a word $w \in X^{\pm 1}$ representing an element of the free group $F(X)$ there is a (usually non-unique) sequence $w=w_0 \to w_1 \to \cdots \to w_r$ with $|w_i|>|w_{i+1}|$ where $w_r$ is the ...
- 1,033
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230
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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 "...
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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 ...
- 243
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1
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554
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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]\...
- 257
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218
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Existence of properly discontinuous and cocompact action
Let $M$ be a complete Riemannian manifold. My question is, under what conditions does $M$ admit a discrete group of isometries $\Gamma$ which acts properly discontinuously and cocompactly on $M$, that ...
- 81
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3
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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
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220
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Carne-Varopoulos bound and stationary measure
Let $\Gamma$ denote the Cayley graph for a finitely generated group $G$, and let $p_n(x, y)$ denote the transition probability that a random walk starting at $x$ reaches $y$ at time $n$. A famous &...
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393
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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 (...
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433
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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 ...
- 41
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186
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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 ...
- 41
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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 ...
- 1,177
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3
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548
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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 ...
- 1,407
8
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1
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227
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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 ...
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1
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259
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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 ...
- 4,432
1
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210
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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 ...
- 1,407
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200
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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 ...
- 629
5
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175
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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 : \...
- 1,163
2
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0
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110
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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 ...
- 4,432
3
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1
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141
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The stabilizer of a pair of points in the acylindrically hyperbolic group is either finite or virtually cyclic
Given a group $G$, suppose $G$ admits a non-elementary acylindrical action
on a Gromov hyperbolic space $S$.
I heard that stabilizer of a pair of points on $\partial S$ in the acylindrically ...
- 199
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176
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Ping pong with parabolic isometries on Gromov hyperbolic spaces
For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
- 41
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66
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Quasi-isometry of solvable minimax groups
[Edits in brackets]
Consider two finitely generated solvable minimax groups $G_i$ ($i = 1,2$) so that $1 \to N_i \to G_i \to Z_i \to 1$ [splits]
with $N_i$ nilpotent, $Z_i$ infinite cyclic and $G_i$ ...
- 4,432
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Characterization of Freudenthal (end) compactification
I had seen somewhere in the literature that the Freudenthal compactification of a locally compact, connected, locally connected, $\sigma$-compact, Hausdorff topological space $X$, is the maximal ideal ...
- 313
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2k
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Bass-Serre theory textbook
I am a PhD freshman working on topological graph theory and geometric group theory. I would like to learn some Bass-Serre theory. What do you think is the best introductory textbook in this topic? ...
- 422
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602
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hyperbolic quotient of hyperbolic group
I have a memory of hearing about a result (or perhaps a conjecture), possibly due to Gromov, that, if $G$ is a hyperbolic group and $g \in G$ has infinite order, then the quotient group $G/\langle (g^...
- 37.4k
17
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1
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459
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Existence of a quasi-isometric residually finite group?
It's, by now, more or less well known that residual finiteness is not a quasi-isometry invariant for finitely generated groups (see here for an example). Thus the following question makes sense:
...
- 733
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345
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Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
- 1,996
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124
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Abelian-by-cyclic subgroups of exponential growth solvable groups
I am currently looking for a reference to a proof (or counterexample) to the following statement:
Statement: Assume $G$ is a finitely generated solvable group of exponential growth, then there is a ...
- 4,432
9
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1
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738
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Gromov hyperbolic groups which are solvable are elementary
I have read on wikipedia that a Gromov hyperbolic group which is solvable is elementary (i.e. virtually cyclic). Where can I find a proof of this fact?
There is a proof of a similar fact in Bridson-...
- 291
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896
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Two definitions of horofunction for Gromov hyperbolic spaces
Let $X$ be a proper, geodesic, $\delta$-hyperbolic metric space (e.g. a hyperbolic group), and let $x_0$ be a basepoint for $X$. There seem to be two different definitions of "horofunction" for $X$, ...
- 8,493
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136
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Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$
Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...
- 1,058
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228
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Automorphism groups of cocompact Fuchsian groups as mapping class groups
Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$
for some $...
- 8,401
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2
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2k
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Modern references on hyperbolic groups
Several good references dedicated to hyperbolic groups have been written until 1990, including:
Hyperbolic groups, written by M. Gromov.
Géométrie et théorie des groupes : les groupes hyperboliques ...
- 8,401
11
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379
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Amalgamated product of automatic groups
In Gersten's "Problems on Automatic Groups", Problem 14, he asks the following question: Let $G=A\ast_{C}B$ where $A$ and $B$ are automatic and $C$ is infinite cyclic. Is $G$ automatic?
Is this ...
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