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      All Questions

      5
      votes
      1answer
      155 views

      “Dimension” of discrete subgroups of infinite covolume in Lie groups

      Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is ...
      3
      votes
      0answers
      103 views

      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 ...
      2
      votes
      1answer
      76 views

      A Backtrack as a Single Word in a Group Presentation yields a Complex that isn't of the Same Homotopy Type?

      By "backtrack" I mean a subword of a relator in a group presentation of the form $x x^{-1}$. Let $X = \langle a \rangle$ as a presentation complex. Let $Y = \langle a$ | $aa^{-1} \rangle$ as a ...
      4
      votes
      0answers
      66 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 ...
      14
      votes
      0answers
      209 views

      What is the centralizer of a Coxeter element?

      Let $(W,S)$ be a Coxeter system (of finite rank) and $c \in W$ a Coxeter element. If $W$ is finite, then the centralizer $C_W(c)$ is the cyclic group generated by $c$ (e.g. see the book "Reflection ...
      4
      votes
      1answer
      119 views

      non-proper parabolic isometries of hyperbolic spaces

      In his seminal paper on hyperbolic groups (see Section 8.1) Gromov defines an isometry $f$ of a hyperbolic space $X$ to be parabolic if the orbit of any point $x\in X$ under the action of $\langle f\...
      4
      votes
      2answers
      205 views

      Is the *unreduced* Burau representation unitary?

      In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...
      7
      votes
      0answers
      175 views

      Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

      Question: Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe ...
      8
      votes
      2answers
      325 views

      Hyperbolic $3$-manifold groups that embed in compact Lie groups

      Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group? It is known that every surface group can be embedded into any semisimple ...
      6
      votes
      2answers
      176 views

      Generalization of Bieberbach's second theorem

      Let $F_0$ and $F_1$ be compact flat manifolds of dimensions $k$ and $m$, respectively, where $k \geq m$. Suppose $f : \pi_1(F_0) \to \pi_1(F_1)$ is a surjective homomorphism. Consider the covering ...
      31
      votes
      2answers
      608 views

      Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

      We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
      6
      votes
      0answers
      115 views

      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 $...
      12
      votes
      1answer
      227 views

      Equivalence of surjections from a surface group to a free group

      Let $g \geq 2$. Let $S = \langle a_1,b_2,...,a_g,b_g | [a_1,b_1] \cdots [a_g,b_g] \rangle$ be the fundamental group of a genus $g$ surface and let $F_g$ be a free group with $g$ generators. Given ...
      4
      votes
      0answers
      161 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, ...
      5
      votes
      0answers
      130 views

      The preimage of bounded real intervals under homomorphisms on hyperbolic groups

      Let $G$ be a hyperbolic group with a fixed (finite, symmetric) generating set and suppose that $\varphi : G \to \mathbb{R}$ is a group homomorphism. Write $W_n = \{ g \in G: |g|=n\}$, where $|g|$ ...

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