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zassenhaus:zassenhaus_2025:program [2025/01/28 16:24] – danielzassenhaus:zassenhaus_2025:program [2025/05/22 13:42] (current) – external edit 127.0.0.1
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 +
 +<div id="header">
 +<h1 style="margin-top:1; margin-bottom:1;">2025 Zassenhaus Groups and Friends Conference</h1></div>
 +<div id="menu">
 +<ul>
 +<li><a href="http://www2.math.binghamton.edu/p/zassenhaus/zassenhaus_2025/home">Home</a></li>
 +<li><a href="http://www2.math.binghamton.edu/p/zassenhaus/zassenhaus_2025/registration">Registration</a></li>
 +<li><a href="http://www2.math.binghamton.edu/p/zassenhaus/zassenhaus_2025/travel_lodging">Travel & Lodging</a></li>
 +<li><a href="http://www2.math.binghamton.edu/p/zassenhaus/zassenhaus_2025/program">Program</a></li>
 +<li><a href="http://www2.math.binghamton.edu/p/zassenhaus/archive">History</a></li>
 +
 +</ul>
 +</div>
 +
 +<br>
 +
 +<section class="timetable" id="schedule">
 +<h2>Conference Schedule</h2>
 +<p>
 +  A detailed program (in PDF format) is
 +  available <a href="https://www2.math.binghamton.edu/lib/exe/fetch.php/zassenhaus/zassenhaus_2025/2025_zassenhaus_program-schedule.pdf">here</a>
 +  <br>
 +  Clicking on a name will take you to the 
 +  talk's title below, and clicking on the title will display its abstract.
 +  
 +<table>
 +  <tbody><tr>
 +      <th>Time</th>
 +      <th>Saturday 5/31</th>
 +      <th>Sunday 6/1</th>
 +</tr>
 +<tr>
 +  <td>8:00</td>
 +  <td class="break" rowspan="2"> Registration</td>
 +  <td></td>
 +</tr>
 +<tr>
 +  <td>8:30</td>
 +  <td><a href="#Riedl"
 + style="color:black"><b>Riedl</b></a></td>
 +  </tr>
 +<tr>
 +    <td>9:00</td>
 +  <td><a href="#Guzman" style="color:black"><b>Guzman</b></a></td>
 +    <td><a href="#McCulloch"
 +    style="color:black"><b>McCulloch</b></a></td>
 +</tr>
 +
 +<tr>
 +  <td>9:30</td>
 +  <td><a href="#Donoven" style="color:black"><b>Donoven</b></a></td>
 +  <td><a href="#Cinarci" style="color:black"><b>&Ccedil;&imath;narc&imath;</b></a></td>
 +</tr>
 +
 +    <tr>
 +  <td>10:00</td>
 +  <td><a href="#Tran" style="color:black"><b>Tran</b></a></td>
 +  <td class="break">Coffee Break</td>
 +</tr>
 +
 +    <tr>
 +      <td>10:30</td>
 +      <td class="break">Coffee Break</td>
 +      <td><a href="#Zarrin" style="color:black"><b>Zarrin</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>11:00</td>
 +      <td><a href="#Lewis" style="color:black"><b>Lewis</b></a></td>
 +      <td><a href="#Summers"
 +      style="color:black"><b>Summers</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>11:30</td>
 +      <td><a href="#Kirtland"
 +      style="color:black"><b>Kirtland</b></a></td>
 +      <td><a href="#Kappe" style="color:black"><b>Kappe</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>12:00</td>
 +      <td><a href="#Klepadlo"
 +      style="color:black"><b>Klepadlo</b></a></td>
 +      <td class="break">Closing Remarks</td>
 +    </tr>
 +    
 +
 +    <tr>
 +      <td>12:20</td>
 +<td class="break">Conference photograph</td>
 +</tr>
 +
 +    <tr>
 +      <td>12:30</td>
 +    <td class="invited" rowspan="3">Lunch Break</td>
 +</tr>
 +    <tr><td>1:00</td>
 +    </tr>
 +    <tr><td>1:30</td></tr>
 +    
 +    <tr>
 +      <td>2:00</td>
 +      <td><a href="#Foguel" style="color:black"><b>Foguel</b></a></td>
 +    </tr>
 +    
 +    <tr><td>2:30</td>
 +      <td><a href="#Russell"
 +      style="color:black"><b>Russell</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>3:00</td>
 +      <td><a href="#Beike" style="color:black"><b>Beike</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>3:30</td>
 +      <td style="break">Coffee Break</td>
 +    </tr>
 +
 +    <tr>
 +      <td>4:00</td>
 +      <td><a href="#Feldman"
 +      style="color:black"><b>Feldman</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>4:30</td>
 +      <td><a href="#Martin" style="color:black"><b>Martin</b></a></td>
 +    </tr>
 +
 +    <tr>
 +      <td>5:00</td>
 +      <td><a href="#Zaremsky"
 +      style="color:black"><b>Zaremsky</b></a></td>
 +    </tr>
 +</tbody></table></div>
 +</section>
 +
 +<section class="abstracts">
 +  <h2>Abstracts</h2>
 +  <div class="talk" id="Guzman">
 +    <p class="talk-title toggle"><b>The isomorphism theorems for
 +      pointed <i>g</i>-digroups</b></p>
 +    <p>Fernando Guzman (Binghamton University) Saturday 9:00 am</p>
 +    <p class="talk-abstract hidden">
 +Digroups, and generalized digroups, <i>g</i>-digroups for short,
 +have been considered as a generalization of continuous groups whose
 +tangent space is a Leibniz algebra. This structure has been seen as a
 +generalization of groups, therefore, efforts have been done to study
 +properties and results that come from group theory, to explore if they
 +hold in this new setting.  A pointed <i>g</i>-digroup is
 +a <i>g</i>-digroup with a 
 +distinguished bar-unit.<br><br>
 +In this talk, we'll discuss the isomorphism theorems for pointed
 +<i>g</i>-digroups, and show that the results for groups do extend to pointed
 +<i>g</i>-digroups.  Most of them also hold for <i>g</i>-digroups.
 +This is joint 
 +work with Olga Patricia Salazar-Diaz.</p>
 +</div>
 +
 +  <div class="talk" id="Donoven">
 +    <p class="talk-title toggle"><b>Generation of simple vigorous
 + groups of homeomorphisms</b></p>
 +    <p>Casey Donoven (Montana State University Northern) Saturday 9:30 am</p>
 +    <p class="talk-abstract hidden">
 +      A group of homeomorphisms <i>G</i> of Cantor space is vigorous if for any
 +      clopen subsets <i>B,C</i>&sub;<i>A</i> of Cantor space,
 +there exists a
 +&gamma;&isin;<i>G</i>
 +such that <i>B</i>&gamma;&sube;<i>C</i>.  Bleak, Hyde, and Elliot
 +(2024) proved that every finitely generated simple vigorous group is
 +2-generated.  In this talk, I will present recent findings extending
 +these results.  For example, if <i>G</i> is a f.g. simple vigorous group,
 +then  (i) <i>G</i> is generated by 3 involutions;
 +(ii) <i>G</i> is generated by an element of order <i>m</i> and <i>n</i> for all
 +<i>m</i>&ge;2 and <i>n</i>&ge;3, (iii)
 +<i>G</i> has a minimal generating set of size <i>k</i> for
 +all <i>k</i>&ge; 2, and (iv)
 +every nontrivial element of <i>G</i> is contained in a generating pair.
 +This is joint work with Collin Bleak, Scott Harper, and James
 +Hyde.</p>
 +    </div>
 +
 +  <div class="talk" id="Tran">
 +    <p class="talk-title toggle"><b>Algebraic Combinatorics meets
 + Probability Theory: Vines and MAT-labeled graphs</b></p>
 +    <p>Tan Tran (Binghamton University) Saturday 10:00 am</p>
 +    <p class="talk-abstract hidden">
 +This talk explores the connection between two concepts from distinct
 +areas of mathematics. The first concept, a vine, is a graphical model
 +used to represent dependent random variables. Initially introduced by
 +Joe (1994) and later formalized by Cooke (1997), vines have become an
 +active research area with applications in probability theory and
 +uncertainty analysis. The second concept, MAT-freeness, is a
 +combinatorial property in the theory of freeness of the logarithmic
 +derivation module of hyperplane arrangements. First studied by
 +Abe-Barakat-Cuntz-Hoge-Terao (2016) and further developed by
 +Cuntz-Muecksch (2020), MAT-freeness has been a topic of increasing
 +interest. In particular, for graphic arrangements, Tsujie and I
 +recently demonstrated that MAT-freeness is completely characterized by
 +the existence of certain edge-labeled graphs, known as MAT-labeled
 +graphs. I will show that there is a fascinating equivalence between
 +the categories of locally regular vines and MAT-labeled
 +graphs. Notably, this leads to an equivalence between the categories
 +of regular vines and MAT-labeled complete graphs. This work is joint
 +with H.M. Tran (Hanoi) and S. Tsujie (Hokkaido). </p>
 +  </div>
 +
 +  <div class="talk" id="Lewis">
 +    <p class="talk-title toggle"><b>Self-normalizing subgroups</b></p>
 +    <p>Mark Lewis (Kent State University) Saturday 11:00 am</p>
 +    <p class="talk-abstract hidden">
 +We consider groups with few conjugacy classes of self--normalizing
 +subgroups. 
 +    </p>
 +  </div>
 +
 +  <div class="talk" id="Kirtland">
 +    <p class="talk-title toggle"><b>2-covering numbers of finite
 + groups</b></p>
 +    <p>Joe Kirtland (Marist University) Saturday 11:30 am</p>
 +    <p class="talk-abstract hidden">
 +A set of proper subgroups is a covering for a group <i>G</i> if its union
 +is the whole group. The minimal number of subgroups needed to cover
 +<i>G</i> is called the covering number of <i>G</i> and is denoted by
 +&sigma;<i>(G)</i>.   A study of coverings of the Paige loop motivates the
 +concept of a 2-covering for a group <i>G</i>, which is a set of proper
 +subgroups of <i>G</i> such that every pair of elements of <i>G</i> are
 +contained 
 +in at least one subgroup in the set.  The minimal number of subgroups
 +needed to 2-cover a group <i>G</i> is called the 2-covering number and
 +denoted by &sigma;<sub>2</sub><i>(G)</i>. Properties of 2-covering
 +numbers will be 
 +presented with the 2-covering number determined for finite nilpotent
 +groups, finite almost simple groups, and particular classes of finite
 +solvable groups. 
 +</p></div>
 +
 +  <div class="talk" id="Klepadlo">
 +    <p class="talk-title toggle"><b>Coverings of dihedral and
 + permutation groups using centralizers</b></a>
 +<p>Matthew Klepadlo (Adelphi University) Saturday 12:00</p>
 +<p class="talk-abstract hidden">
 +A group is said to be covered if there exists proper subgroups such
 +that their union is the same as the whole group. This paper will go
 +into how we use centralizer subgroups to come up with coverings of
 +smaller dihedral and permutation groups and obtain the "covering
 +number" and "centralizer-covering number." We will also be
 +highlighting a few notable theorems regarding coverings and use them
 +to our advantage to finding said "covering/centralizer-covering
 +number." The history of coverings and the interest mathematicians have
 +in them will also be explored. </p></div>
 +
 +<div class="talk" id="Foguel">
 +  <p  class="talk-title toggle"><b>Finite groups in which every
 +      subgroup of order divisible by <i>p</i> is normal</b></p>
 +  <p>Tuval Foguel (Adelphi University) 2:00 pm</p>
 +  <p class="talk-abstract hidden">
 +In this talk, I’ll introduce two generalizations of Dedekind groups,
 +called <i>PN</i>-groups and <i>PNQ</i>-groups. In <i>PN</i>-groups,
 +every subgroup 
 +whose order is divisible by a fixed prime <i>p</i> is normal, while in
 +<i>PNQ</i>-groups, such subgroups are permutable. We’ll begin by showing
 +that these groups must be either <i>p</i>'-groups or
 +supersolvable. From there, I’ll walk through a classification of both
 +<i>PN</i>- and <i>PNQ</i>-groups. I’ll end with a brief discussion of minimal
 +non-<i>PN</i>-groups and some questions that remain open. </p></div>
 +
 +<div class="talk" id="Russell">
 +  <p class="talk-title toggle"><b>Extended Springer fibers
 +      overview</b></p>
 +  <p>Amber Russell (Butler University) Saturday 2:30 pm</p>
 +  <p class="talk-abstract hidden">
 +The Springer Correspondence associates to each irreducible
 +representation of the Weyl group for a reductive Lie algebra a
 +nilpotent orbit for the Lie algebra and an irreducible representation
 +of the fundamental group of the Lie algebra.  This result was due to
 +T.A. Springer in the 1970s and is still providing fertile grounds of
 +innovation today.  The key tools in this work were the Springer
 +resolution, a resolution of singularities for the nilpotent cone of
 +the Lie algebra, and also careful study of the resulting Springer
 +fibers. In the 1980s, George Lusztig expanded this to a bijection
 +where all possible pairs of nilpotent orbits and irreducible
 +representations of the fundamental group appear and the Weyl group is
 +replaced by a class of new relative Weyl groups.  This is Lusztig's
 +Generalized Springer Correspondence.<br><br>
 +Over the past several years, I have collaborated will William Graham
 +and Martha Precup on a related project, spanning multiple publications
 +with a new one currently being prepared.  In particular, we have
 +studied Extended Springer Fibers and connected them to Lusztig's
 +Generalized Springer Correspondence in all classical types and
 +relevant exceptional types.  The goal of this presentation will be an
 +overview of these results. 
 +</p></div>
 +
 +<div class="talk" id="Beike">
 +  <p class="talk-title toggle"><b><i>p</i>-Groups with derived length
 +      three and three character degrees</b></p>
 +<p>Nic Beike (Kent State University) Saturday 3:00 pm</p>
 +  <p class="talk-abstract hidden">
 +We will construct examples of <i>p</i>-groups with derived length 3 and 3
 +character degrees. We will focus on groups of order <i>p</i><sup>6</sup>. 
 +</p></div>
 +
 +<div class="talk" id="Feldman">
 +  <p class="talk-title toggle"><b>Another dual to Schunck
 +      classes</b></p>
 +  <p>Arnold Feldman (Franklin and Marshall College) Saturday 4:00
 +    pm</p>
 +  <p class="talk-abstract hidden">
 +The duality of Fitting classes and formations has been widely studied,
 +and Fitting classes and Schunck classes can also be considered
 +to be dual in some sense. However, the definitions of Fitting
 +and Schunck classes are not literally dual the way that those of
 +Fitting classes and formations are.  Here we identify a dual to
 +Schunck classes of finite groups, which we call SchunckD
 +classes, based on the standard definition of a Schunck class.
 +We investigate properties and examples of Schunck classes and
 +see how they differ from Fitting classes.  The arguments are
 +relatively elementary, and the topic could lend itself to
 +exploration by advanced undergraduates.
 +</p></div>
 +
 +<div class="talk" id="Martin">
 +  <p class="talk-title toggle"><b>Groups with a fixed character
 +      degree</b></p>
 +  <p>Brandon Martin (Kent State University) Saturday 4:30 pm</p>
 +  <p class="talk-abstract hidden">
 +Let <i>x=d<sub>1</sub>&hellip;d<sub>
 +      m </sub>p<sub>1</sub><sup>a<sub>1</sub></sup> &hellip; p<sub>n</sub><sup>a<sub>n</sub></sup></i>
 +where the <i>d<sub>j</sub></i>'s and <i>p<sub>i</sub></i>'s are
 +      distinct primes, and <i>a<sub>i</sub></i>&isin;<b>N</b>
 +for all <i>i</i>. Let 
 +<i>d=d<sub>1</sub>&hellip;d<sub>m</sub></i>.  We've previously shown there
 +exists a solvable 
 +group <i>G</i>, of order <i>x</i>, with <i>d</i>&isin;cd<i>(G)</i> if
 +and only if there 
 +is a sequence of congruences between the <i>p<sub>i</sub></i>'s and <i>d<sub>j</sub></i>'s where the
 +product of the moduli of these congruences is precisely <i>d</i>.  Now, we
 +let relax the square-free condition on <i>d</i> and consider when an
 +analogous result holds when <i>x=d<sup>m</sup>p<sup>n</sup></i>. 
 +</p></div>
 +
 +<div class="talk" id="Zaremsky">
 +  <p class="talk-title toggle"><b>Some difficult simple groups</b></p>
 +  <p>Matt Zaremsky (University at Albany) Saturday 5:00 pm</p>
 +  <p class="talk-abstract hidden">
 +Finite simple groups are famously classified, but infinite simple
 +groups remain extremely mysterious in general. In particular, a famous
 +conjecture of Boone and Higman predicts that every finitely generated
 +group with solvable word problem embeds in a finitely presented simple
 +group, so finitely presented simple groups are conjecturally
 +ubiquitous, but actual examples are hard to come by. In this talk I
 +will survey some of the bizarre and interesting (infinite) simple
 +groups that arise, and mention some recent results, with a focus on a
 +family of simple groups called twisted Brin-Thompson
 +groups. </p></div>
 +
 +<div class="talk" id="Riedl">
 +  <p class="talk-title toggle"><b>Images of iterated commutators under
 +      group automorphisms</b></a>
 +<p>Jeffrey Riedl (University of Akron) Sunday 8:30 am</p>
 +<p class="talk-abstract hidden">
 +Let <i>x</i>, <i>y</i> be elements of a group <i>G</i>. For each
 +integer <i>m</i>&ge;0 let 
 +<i>d<sub>m</sub></i> denote the <i>m</i>th iterated commutator of <i>x</i>
 +  by <i>y</i>. Thus
 +<i>d<sub>0</sub>=x</i>, <i>d<sub>1</sub>=[x,y]</i>, <i>d<sub>2</sub>=[[x,y],y]</i>,
 +and so on. Let <b>D</b><i>=
 +={d<sub>0</sub>,d<sub>1</sub>, d<sub>2</sub>,&hellip;}</i> and suppose
 +all the elements of <b>D</b>
 +commute with each other. Let &sigma; be an automorphism of <i>G</i> such
 +that <i>x<sup>&sigma;</sup>=x</i>
 +and <i>y<sup>&sigma;</sup>=y<sup>u</sup></i> for some positive integer 
 +<i>u</i>. We establish a formula that expresses the image under &sigma; of
 +an arbitrary element of <b>D</b> as a product of elements of <b>D</b>.
 +<br><br>
 +We mention the application that motivated the establishment of this
 +formula. Let <i>p</i> be a prime and let <i>n</i>&ge;2. The group
 +U(<i>p<sup>n</sub></i>) of
 +units modulo <i>p<sup>n</sup></i> acts naturally via automorphisms on
 +the regular 
 +wreath product group <i>W=Z<sub>p<sup>2</sup></sub></i> wr
 +  <i>Z<sub>p</sup>n</sup></sub></i>, and hence acts on the set
 +<b>N</b> consisting of all the normal subgroups of <i>W</i> that are
 +contained in the base group of <i>W</i>. The formula enables the
 +straightforward computation of the image <b>N</b><sup>&sigma;</sup> of
 +an arbitrary 
 +<i>N</i>&isin;<b>N</b> for an arbitrary automorphism &sigma;&isin;
 +U(<i>p<sup>n</sup></i>).  
 +</p></div>
 +
 +<div class="talk" id="McCulloch">
 +  <p class="talk-title toggle"><b>Groups with dense Chermak-Delgado
 +      subgroups</b></p>
 +  <p>Ryan McCulloch (Binghamton University) Sunday 9:00 am</p>
 +  <p class="talk-abstract hidden">
 +Let <b><i>X</i></b> be a property pertaining to subgroups of a group. We
 +say that a group 
 +<i>G</i> has dense <b><i>X</i></b>-subgroups if for each pair <i>(H, K)</i> of
 +subgroups of <i>G</i> such that <i>H</i> &lt;<i>K</i> and <i>H</i> is
 +not maximal in <i>K</i>, 
 +there exists an <b><i>X</i></b>-subgroup <i>X</i> of <i>G</i> such
 +that <i>H &lt; X &lt;  
 +K</i>.  In this talk we consider groups with Chermak--Delgado dense
 +subgroups and, more generally, with centralizer dense subgroups.  This
 +includes joint work with Marius Tarnauceanu.    </p></div>
 +
 +<div class="talk" id="Cinarci">
 +  <p class="talk-title toggle"><b>Some results on derived length and
 +      character degrees</b></p>
 +  <p>Burcu &Ccedil;&imath;narc&imath; (Texas State University)
 +    Sunday 9:30 am</p>
 +  <p class="talk-abstract hidden">
 +The character degrees of a finite group provide some important
 +information about the structure of the group. A famous problem on the
 +character degrees of a finite solvable group <i>G</i> is known as the
 +Taketa problem and Isaacs-Seitz conjecture. This problem states that
 +the inequality <i>dl(G) &le; |cd(G)|</i> holds for a finite solvable group
 +<i>G</i>, where <i>dl(G)</i> is the derived length of <i>G</i> and
 +<i>|cd(G)|</i> is the
 +cardinality of the set of all irreducible character degrees of
 +<i>G</i>. Although this conjecture is still open, many research articles
 +have been published on this inequality. In this talk, we show that the
 +Taketa inequality holds for <i>G</i> under some sufficient conditions.
 +</p></div>
 +
 +<div class="talk" id="Zarrin">
 +  <p class="talk-title toggle"><b>On the noncommuting set in infinite
 +      groups</b></p>
 +  <p>Mohammad Zarrin (Texas State University) Sunday 10:30 am</p>
 +  <p class="talk-abstract hidden">
 +Let <i>G</i> be a non-abelian group. A subset <i>T</i> of a
 +group <i>G</i> is a set of 
 +pairwise noncommuting elements if <i>xy&neq; yx</i> for any two distinct
 +elements <i>x</i> and <i>y</i> in <i>T</i>.<br><br>
 +If <i>|T| &ge; |R|</i> for any other set of pairwise noncommuting elements
 +<i>R</i> in <i>G</i>, then <i>T</i> is called a maximal subset of pairwise
 +noncommuting elements and the cardinality of such a subset (if it
 +exists) is denoted by <i>w(G)</i>. In this talk, among other things, we
 +show that, for each positive integer <i>m</i>, there are only finitely many
 +groups <i>G</i>, up to isoclinism, with <i>w(G) = m</i>, and we obtain similar
 +results for groups with exactly <i>m</i> centralizers.<br><br>
 +Also, we try to find the influence of the function <i>w(G)</i> on the
 +structure of groups. </p></div>
 +
 +<div class="talk" id="Summers">
 +  <p class="talk-title toggle"><b>On the number of disconnected
 +      character degree graphs satisfying P&aacute;lfy's inequality</b></p>
 +  <p>Andrew Summers (Kent State University) Sunday 11:00 am</p>
 +  <p class="talk-abstract hidden">
 +Let <i>G</i> be a finite solvable group with disconnected character degree
 +graph &Delta;(<i>G</i>). Under these conditions, it follows from a result of
 +P&aacute;lfy that &Delta;(<i>G</i>) consists of two connected
 +components. Another 
 +result of P&aacute;lfy's gives an inequality relating the sizes of these two
 +connected components. In this talk, some background on character
 +degree graphs and P&aacute;lfy's results will be presented. The number of
 +possible component size pairs that satisfy Pálfy's inequality will be
 +calculated. Additionally, for a fixed positive integer <i>n</i>, the number
 +of distinct graph orders for which exactly <i>n</i> component size pairs
 +satisfy P&aacute;lfy's inequality is shown. 
 +</p></div>
 +
 +<div class="talk" id="Kappe">
 +  <p class="talk-title toggle"><b>Element centralizers in a group
 +      centralizer lattice and centralizer-like subgroups</b></p>
 +  <p>Luise-Charlotte Kappe (Binghamton University) Sunday 11:30 am</p>
 +  <p class="talk-abstract hidden">
 +We note some properties of the centralizer map and recall the
 +centralizer lattice of a group. Since the element centralizers
 +generate all the other centralizers, we consider how the element
 +centralizers sit in the lattice. We generalize this by considering the
 +so-called centralizer-like subgroups of a group associated with a
 +2-letter word <i>w(u,v)</i>. These are four subgroups defined by an
 +operator that takes as input a subgroup <i>H</i> and returns the subgroup
 +of group elements <i>x</i> such
 +that <i>w(xg,h)=w(g,h)</i>, <i>w(gx,h)=w(g,h)</i>, 
 +<i>w(h,xg)=w(h,g)</i>, and <i>w(h,gx) = w(h,g)</i> respectively, for
 +all <i>g</i>&isin;<i>G</i> and <i>h</i>&isin;<i>H</i>. We investigate
 +for which words these centralizer-like 
 +subgroups also generate a lattice that is a centralizer-like lattice.
 +<br><br>
 +This is joint work with Wil Cocke, Mark Lewis, and Ryan McCulloch.
 +</p></div>    
 +
 +</section>
 +
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