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show/hide this revision's text 3 Added Gerhard's example; added 30 characters in body

There is a natural topology on the automorphism group here, as in your other question, whose basic open sets are determined by a finite piece of the automorphism. I expect that one can put Indeed, the group is essentially the same as Cantor space $2^{\mathbb{N}}$, since there is a nice one-to-one correspondence between the binary sequences and the automorphisms, just by considering the diamonds in which the swap is made or not. This space has a natural probability measure, treating the swaps as coin flips. Thus, the collection of automorphisms that make $n$ many prescribed swaps or non-swaps has measure $2^{-n}$. This measure interacts well with the topology, because they are both the standard concepts on itCantor space.

Finally, to address Gerhard's idea of building a poset satisfying the properties but I'd having a countably infinite automorphism group, while having diamonds, let us simply join the diamonds horizontally:

* * * * / \ / \ / \ / \ ... -2 -1 0 1 2 ... \ / \ / \ / \ / * * * *

This poset is locally finite, countable and connected, and it exhibits the extra property because every point has only finitely many successors and predecessors. Meanwhile, the automorphism group is countably infinite because it is precisely the infinite dihedral group: we have to think more about exactly the precise detailshorizontal translations and reflections. (And to satisfy Gerhard, we have many diamonds!)
show/hide this revision's text 2 added 29 characters in body

The answer to

For question 1is no, the automorphism group needn't be countable. To see this, consider the poset consisting of infinitely many diamonds stacked on top of one another. 

     .
     .
     .

     3
    / \
   *   *
    \ /
     2
    / \
   *   *
    \ /
     1
    / \
   *   *
    \ /
     0

This is locally finite, countable and connected in the sense you have requested. Your additional property is satisfied because every antichain (in the sense of pairwise incomparable elements) has size at most two. The automorphism group is uncountable, however, because we can swap the two intermediate points of any diamond independently, and so any infinite binary sequence determines a distinct manner of performing these swaps.

There is a natural topology on the automorphism group here, as in your other question, whose basic open sets are determined by a finite piece of the automorphism. I expect that one can put a nice measure on it, but I'd have to think more about the precise details.

show/hide this revision's text 1

The answer to question 1 is no. To see this, consider the poset consisting of infinitely many diamonds stacked on top of one another. 

     .
     .
     .

     3
    / \
   *   *
    \ /
     2
    / \
   *   *
    \ /
     1
    / \
   *   *
    \ /
     0

This is locally finite, countable and connected in the sense you have requested. Your additional property is satisfied because every antichain (in the sense of pairwise incomparable elements) has size at most two. The automorphism group is uncountable, however, because we can swap the two intermediate points of any diamond independently, and so any infinite binary sequence determines a distinct manner of performing these swaps.

There is a natural topology on the automorphism group here, as in your other question, whose basic open sets are determined by a finite piece of the automorphism. I expect that one can put a nice measure on it, but I'd have to think more about the precise details.