scholarly journals Nielsen realization for infinite-type surfaces

2020 ◽  
pp. 1
Author(s):  
Santana Afton ◽  
Danny Calegari ◽  
Lvzhou Chen ◽  
Rylee Lyman
Keyword(s):  
Infinite Type ◽  
Nielsen Realization

scholarly journals Random additions in urns of integers

2021 ◽  
Vol 58 (2) ◽  
pp. 335-346
Author(s):  
Mackenzie Simper
Keyword(s):  
Finite Group ◽  
Random Process ◽  
Uniform Distribution ◽  
Discrete Time ◽  
Random Variable ◽  
Urn Model ◽  
Infinite Type ◽  
The Mean ◽  
A Finite Group ◽  
Proof Of Convergence

AbstractConsider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn labels. This model was introduced by Siegmund and Yakir (2005) Ann. Prob.33, 2036 for labels taking values in a finite group, in which case the distribution defined by the urn converges to the uniform distribution on the group. For the urn of integers, the main result of this paper is an exponential limit law. The mean of the exponential is a random variable with distribution depending on the starting configuration. This is a novel urn model which combines multi-drawing and an infinite type of balls. The proof of convergence uses the contraction method for recursive distributional equations.

scholarly journals Alvis–Curtis duality for representations of reductive groups with Frobenius maps

2020 ◽  
Vol 32 (5) ◽  
pp. 1289-1296
Author(s):  
Junbin Dong
Keyword(s):  
Irreducible Representations ◽  
Reductive Groups ◽  
Infinite Type ◽  
Abstract Representations

AbstractWe generalize the Alvis–Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis–Curtis duality of infinite type, which we define in this paper, also interchanges the irreducible representations in the principal representation category.

On a conjecture in Artin groups

2021 ◽  
pp. 1-25
Author(s):  
Arye Juhász
Keyword(s):  
Artin Group ◽  
Two Dimensional ◽  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Infinite Type ◽  
Trivial Center

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

Circle homeomorphism with infinite type of singularities

2015 ◽  
Author(s):  
Sokhobiddin Akhatkulov ◽  
Mohd. Salmi Md. Noorani
Keyword(s):  
Circle Homeomorphism ◽  
Infinite Type

scholarly journals The probabilities of extinction in a branching random walk on a strip

2020 ◽  
Vol 57 (3) ◽  
pp. 811-831
Author(s):  
Peter Braunsteins ◽  
Sophie Hautphenne
Keyword(s):  
Random Walk ◽  
Iterative Method ◽  
Fixed Points ◽  
Branching Processes ◽  
Extinction Probability ◽  
Branching Random Walk ◽  
Infinite Type ◽  
Hessenberg Form ◽  
Global Extinction ◽  
Countably Infinite

AbstractWe consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

scholarly journals Non-realizability of the pure braid group as area-preserving homeomorphisms

2020 ◽  
pp. 1-12
Author(s):  
LEI CHEN
Keyword(s):  
Braid Group ◽  
Natural Projection ◽  
Realization Problem ◽  
Pure Braid Group ◽  
Rotation Numbers ◽  
Marked Points ◽  
Area Preserving ◽  
Nielsen Realization

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

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