# Tagged Questions

Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

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### 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 ...
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### How bad is the modular space?

I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$? Do we know something about its homology or homotopy groups ? $\mathbb{H}^{3}$ is the hyperbolic ...
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This is an attempt to find a correct version of Do free profinite groups satisfy Howson's theorem? Let $F$ be a free profinite group, and let $A,B \leq F$ be closed finitely generated subgroups ...
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### Do limit groups satisfy Howson's theorem?

Let $G$ be a limit group, and let $A,B \leq G$ be finitely generated subgroups generating $G$ (i.e. $\langle A \cup B \rangle = G$). Must $A \cap B$ be finitely generated? Recall that a limit ...
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### Homology of spherical braid groups

By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...
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### Conjugacy of matrices of order three in $PGL(2,k)$, where $k$ is any field

We know that every matrix of order three in $PGL(2,\mathbb Z)$ is conjugate to the following matrix $$\left( \begin{array}[cc] &1 & -1 \\ 1&0 \end{array} \right)$$ I want to know if ...
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### Rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$

What is the rational cohomological dimension of the lattices in $SL_n(\mathbb{Q}_p)$, where $n\geq 3$ ? A reference would be appreciated. For the definition of "cohomological dimension of a group ...
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### Quotients of an extension of the Higman group

(Note: this started as a different question that soon changed form, thanks to the answers.) Let $G = \mathbb{Z}/4\mathbb{Z} \ltimes H_4$, where $H_4$ is the Higman group and $\mathbb{Z}/4\mathbb{Z}$ ...
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### Algorithm to generate hyperbolic metric on a compact surface

Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
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### 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 ...
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### A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers

I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that a) the graph $\Gamma$ is fine, b) $\Gamma$ is not a tree, c) not all ...
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### Cubulating non-compact hyperbolic manifolds

Let $X$ be a hyperbolic manifold of arbitrary dimension. When does $X$ admit a cell structure of a CAT(-1) cube complex? of a hyperbolic CAT(0) cube complex? I suspect that the question is widely ...
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### A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that: For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index ...
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### k-fellow traveler property and automatic structur

Let P be a permutation group with some generating set S and let W be the word acceptor automaton of P, if I know the value of k (k-fellow-traveller property of CayleyGraph CG(P,S)). I realized that ...
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### Local quasiconvexity in graphs of free groups with cyclic edge groups

In Wise1 Wise shows that hyperbolic graphs of free groups with cyclic edge groups are subgroup separable. In Hsu-Wise these are shown to be cubulated, and by Agol they're virtually special, so ...
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### Do quasi-isometric groups have the same rational cohomology?

Let $G_1$ and $G_2$ be two finitely generated groups which are quasi-isometric in the sense of geometric group theory. Are their rational cohomology rings $H^{\ast}(G_i; \mathbb Q)$ necessarily ...
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### Computations with conetypes of hyperbolic groups

I'd like to know if there exists (and, in this case, where I can find it) some computer program/programming language/any kind of software that can find explicitly the conetypes of a hyperbolic group ...
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### Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows: Suppose ...
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### 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 ...
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### Is there a one relator group with property (T)?

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 generated by $x$ has Kazhdan's property $\mathrm{(T)}$ ?
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### Detecting HNN-Extension and free products with amalgamation

This question is partly connected with the following Connection between Stalling's end theorem and Seifert-van Kampen Theorem. By Stalling's Theorem a group with more than one end splits over a ...
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### Obtaining a quasi-isometry of the 'boundary'

It is well-known that a quasi-isometry induces a homeomorphism on the space of ends of say a locally finite graph for simplicity. Clearly the converse is not true. In other words the concept of ends ...
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### Connection between Stalling's end theorem and Seifert-van Kampen Theorem

Stalling`s end Theorem (a group has more than one end iff it splits over a finite subgroup) and the Seifert-van Kampen Theorem (the fundamental group of a 'decomposable' space is a free amalgamated ...
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### Is every metric space quasi-isometric to a graph?

I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$...during this proof I've precisely used the fact that given two point in $X$ there ...
Let's say the group is generated by three involutions $a$, $b$, and $c$ such that $ord(ab)= 2$, $ord(bc)=3$, and $ord(ac)=m$. Under which conditions is it isomorphic to the rank 3 Coxeter group $(2, 3,... 0answers 563 views ### Reference request: Parallel processor theorem of William Thurston Sometime in the 1980's or 1990's, Bill Thurston proved a theorem regarding the existence of a universal parallel processing machine, using a certain class for such machines having finite deterministic ... 0answers 162 views ### divisible 2nd cohomolgy group$H^2(G,\mathbb{Z}G)$Recall that a (nontrivial) abelian group$A$is called divisible if the multiplication by$n$is surjection$A\to A$for all$n\in\mathbb{Z}_{>1}$. My question is the following. Is there a ... 0answers 111 views ### Poincaré inequality for connected Lie groups Let$G$be a compactly generated second countable locally compact group, and let$\mu$be a probability measure which is: symmetric, adapted (in the sense that there is no proper subgroup$H$such ... 0answers 69 views ### Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant? Is the family of probabilities$\mu^n$(convolution) generated by a random walk$\mu$on a finitely generated amenable group$G$asymptotically invariant ($\|g\mu^n-\mu\|_{L^1}\to 0$for any$g\in G$)?... 2answers 232 views ### Products of elliptic isometries A well-known property on groups acting on trees is: Theorem: Let$T$be a tree and$g,h \in \mathrm{Isom}(T)$two elliptic isometries. If$\mathrm{Fix}(g) \cap \mathrm{Fix}(h) = \emptyset$then ... 0answers 163 views ### Uniform sub-linearity of sub-additive functions on groups Suppose$G$is a finitely generated group and suppose$f: G \to \mathbb{R}$is subadditive function, that is:$f(g_1\circ g_2) \leq f(g_1) + f(g_2)$. One example of such$f$is the word length in some ... 3answers 750 views ### Your favorite papers on geometric group theory I would like to improve my culture on geometric group theory, so I am interested in short and accessible papers on subjects related to this field but which are not always available in the classical ... 0answers 93 views ### Number of unitary representations of a Kazhdan group It was proved by de la Harpe, Robertson, and Valette that for a discrete group$\Gamma$with Kazhdan's property (T), there is a constant$c$so that the number of irreducible unitary representations ... 1answer 283 views ### On the number of ends of a countable simple group At the beginning I thought that the following statement could be an easy exercise after Stallings' theorem, but I found myself incapable of proving it: Any countable f.g. simple group has one end. ... 1answer 218 views ### Abelianization of limit groups Let$G_1$and$G_2$be limit groups, and let$C_1$and$C_2$be cyclic subgroups of$G_1$and$G_2$, respectively. Question: If$G$is the amalgamated product of$G_1$and$G_2$with amalgamated ... 0answers 141 views ### Hyperbolic manifold of dim 3 with finite volume. The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) ... 1answer 128 views ### A subgroup of corank 1 in a free group contains a primitive element? Let$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 \...
Let $F$ be a free group of rank $\aleph_0$ and let $H$ be a subgroup for which there exists some $a \in F$ such that $\langle H \cup \{a\} \rangle = F$. Must there be some free basis $B$ for $F$ and a ...